# ExoMol line lists – XLVI: Empirical rovibronic spectra of silicon
mononitrate (SiN) covering the 6 lowest electronic states and 4 isotopologues

Mikhail Semenov1, Nicholas Clark1, Sergei N. Yurchenko1, Gap-Sue Kim2,
Jonathan Tennyson1
1 Department of Physics and Astronomy, University College London, Gower
Street, WC1E 6BT London, UK
$2$ Dharma College, Dongguk University, 30, Pildong-ro 1-gil, Jung-gu, Seoul
04620, Korea The corresponding author<EMAIL_ADDRESS>
###### Abstract

Silicon mononitride (28Si14N, 29Si14N, 30Si14N, 28Si15N) line lists covering
infrared, visible and ultraviolet regions are presented. The SiNful line lists
produced by ExoMol include rovibronic transitions between six electronic
states: $X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$, $B\,{}^{2}\Sigma^{+}$,
$D\,{}^{2}\Delta$, $a\,{}^{4}\Sigma^{+}$, $b\,{}^{4}\Pi$. The ab initio
potential energy and coupling curves, computed at the multireference
configuration interaction (MRCI/aug-cc-pVQZ) level of theory, are refined for
the observed states by fitting their analytical representations to 1052
experimentally derived SiN energy levels determined from rovibronic bands
belonging to the $X$–$X$, $A$–$X$ and $B$–$X$ electronic systems through the
MARVEL procedure. The SiNful line lists are compared to previously observed
spectra, recorded and calculated lifetimes, and previously calculated
partition functions. SiNful is available via the www.exomol.com database.

###### keywords:

molecular data, opacity, astronomical data bases: miscellaneous, planets and
satellites: atmospheres, stars: low-mass.

††pagerange: ExoMol line lists – XLVI: Empirical rovibronic spectra of silicon
mononitrate (SiN) covering the 6 lowest electronic states and 4
isotopologues–References

##  1 Introduction

Silicon is considered to be the second most abundant element in Earths crust
(CRC Handbook, 2016) and the seventh most abundant element on the planet
according to McDonough & Sun (1995); low silicon abundance within a host is
considered to be a better indicator of a potential planet detection than
planet-metallicity correlation according to Brugamyer et al. (2011). So far
there have been multiple observations of SiN in different media in space:
interstellar medium (Schilke et al., 2003; Turner & Dalgarno, 1977; Feldman et
al., 1983), red giant stars (Davis, 1947; Gratton, 1952), and envelopes of
carbon stars (Turner, 1992). Apart from being an important astrophysical
species, SiN has multiple different applications: quantum dot production (Xie
et al., 2017), quantum optomechanics (Serra et al., 2018), protective coating
for biological tissues and dental implants (Pettersson et al., 2013; Raza et
al., 2020), membranes used in filtering and biosensor systems (Vlassiouk et
al., 2009; Lee et al., 2014).

The SiN molecule seems to be well studied experimentally in comparison to some
of its valence group counterparts. Originally the molecule was reported by
Jevons (1913), with band heads attributed to the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ band. Later Mulliken (1925)
conducted a more thorough study of the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ reporting rotational transitions
and detecting a weaker band later identified as
$C\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ thanks to additional experimental works
conducted by Linton (1975). Overall the main active electronic bands that have
been rotationally resolved so far are
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ (Ojha & Gopal, 2013; Ito et al.,
1993; Naulin et al., 1993; Piper & Caledonia, 1991; Foster, 1989; Walkup et
al., 1984; Foster, 1984; Bredohl et al., 1976; Singh et al., 1973; Dunn et
al., 1969; Nagaraj & Verma, 1968; Schofield & Broida, 1965; Stevens &
Ferguson, 1963; Jenkins & de Laszlo, 1928),
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ (Meloni et al., 2004; Elhanine et al.,
1992; Foster, 1989; Yamada et al., 1988; Yamada & Hirota, 1985; Foster et al.,
1985; Foster, 1984; Bredohl et al., 1976; Jevons, 1913) and
$C\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ (Foster, 1989; Bredohl et al., 1976;
Linton, 1975; Mulliken, 1925), $B\,{}^{2}\Sigma^{+}$–$A\,{}^{2}\Pi$ (Bredohl
et al., 1976). Of these Ojha & Gopal (2013) is the most recent experimental
study, which apart from the $B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ band
mentioned earlier, has also been able to detect and vibrationally resolve the
$f\,{}^{4}\Sigma^{-}$–$d\,{}^{4}\Sigma^{-}$,
$b\,{}^{4}\Pi$–$d\,{}^{4}\Sigma^{-}$. Additionally we want to highlight that
Nagaraj & Verma (1968) originally assigned their spectrum and quantum numbers
to the SiO+ anion, however, that was later corrected by the Dunn et al. (1969)
and assigned to the $B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ band of SiN.

A large number of theoretical investigations of SiN have been performed since
the discovery of the molecule (Xing et al., 2018, 2013; Oyedepo et al., 2011;
Li et al., 2009; Kerkines & Mavridis, 2005; Jungnickel et al., 2000; Singh et
al., 1999; Cai et al., 1998; Borin, 1996; Chong, 1994; Chen et al., 1993;
Mclean et al., 1992; Curtiss et al., 1991; Melius & Ho, 1991; Muller-Plathe &
Laaksonen, 1989; Ziurys et al., 1984; Bruna et al., 1984; Preuss et al., 1978;
Gohel & Shah, 1975). The most recent ab initio work at the time of writing was
carried out by Xing et al. (2018) who conducted a thorough analysis of the
transition dipole moment curves and potential energy curves for the lowest 8
doublets ($X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$, $B\,{}^{2}\Sigma^{+}$,
$C\,{}^{2}\Pi$, $D\,{}^{2}\Delta$, $E\,{}^{2}\Delta$,$F\,{}^{2}\Pi$,
$G\,{}^{2}\Delta$), of SiN, additionally reporting on Frank-Condon factors and
lifetimes for the first 15 vibrational levels of each electronic state, for
which they used program LEVEL due to Le Roy (1998). The states were studied at
the Internally Contracted Multirefrence Configuration Interaction with
Davidson correction (icMRCI+Q) level with additional extrapolation procedure
employed as described by Oyeyemi et al. (2014). A lot of work in the 2018
paper was built on the foundation of their earlier paper (Xing et al., 2013),
where they reported PECs for the lowest 8 doublets ($X\,{}^{2}\Sigma^{+}$,
$A\,{}^{2}\Pi$, $B\,{}^{2}\Sigma^{+}$, $C\,{}^{2}\Pi$, $D\,{}^{2}\Delta$,
$E\,{}^{2}\Delta$, $F\,{}^{2}\Pi$, $G\,{}^{2}\Delta$) and 4 quartets
($a\,{}^{4}\Sigma^{+}$, $b\,{}^{4}\Pi$, $c\,{}^{4}\Delta$,
$d\,{}^{4}\Sigma^{-}$) and the lowest sextet state $1^{6}\Sigma^{+}$. Apart
form PECs this paper also reported spectroscopic constants and discussed the
effects of the spin-orbit couplings.

The work described below will present original ab initio potential energy
curves (PECs), spin-orbit curves (SOCs), electronic angular momentum curves
(EAMCs) and (transition) dipole moment curves (DMCs) calculated using a high
level of theory with MOLPRO (Werner et al., 2020) and then empirically fitted
using data obtained from a MARVEL (measured active rotation vibration energy
levels) (Furtenbacher et al., 2007) procedure and generated from effective
Hamiltonians using PGOPHER (Western, 2017). The refined spectroscopic model is
then used to compute molecular line lists for four isotopologues of SiN. Both
the refinement and line list calculations are performed using the rovibronic
program Duo (Yurchenko et al., 2016). We then use our line lists to give high
accuracy comparison to previously reported experimental spectra, calculated
and experimental lifetimes, and partition function.

##  2 Ab initio model

In this work six lowest electronic states, namely $X\,{}^{2}\Sigma^{+}$,
$A\,{}^{2}\Pi$, $B\,{}^{2}\Sigma^{+}$, $D\,{}^{2}\Delta$,
$a\,{}^{4}\Sigma^{+}$, $b\,{}^{4}\Pi$, were studied using the state-averaged
complete active space self consistent field (CASSCF) and multireference
configuration interaction (MRCI) methods and aug-cc-pVQZ basis sets. The
calculations were performed using C2v point group symmetry. Potential energy
curves, illustrated in Fig. 1, spin-orbit coupling and electronic angular
moment (Fig. 2), dipole moment curves and transition dipole moment curves
(Fig. 3) were computed using MOLPRO2020 (Werner et al., 2020). If the MOLPRO
calculations at some geometries did not converge, they were interpolated or
extrapolated from the neighbouring points as part of the rovibronic
calculations (see below). We used an adaptive ab initio grid consisting of 121
bond lengths ranging from 1.1 to 3.08 Å with more points around the
equilibrium region of the $X\,{}^{2}\Sigma^{+}$ state centred at 1.585 Å, see
Figs. 1–3, where the density of the ab initio grid is shown. The grid points
with the corresponding ab initio values (if converged) are included in the
supplementary material.

For ab initio calculation we have considered the states of SiN correlating
with two dissociation asymptotes, Si(3P) + N(4S) and Si(3P) + N(2D), which
generate $\,{}^{2}\Sigma^{+}$(three states), $\,{}^{2}\Pi$ (one state),
$\,{}^{4}\Sigma^{+}$(one states) and $\,{}^{4}\Pi$ (one state) which show six
electronic states, $X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$,
$B\,{}^{2}\Sigma^{+}$, $D\,{}^{2}\Delta$, $a\,{}^{4}\Sigma^{+}$ and
$b\,{}^{4}\Pi$ which are mentioned above in ab initio model section. Under the
C2v point group symmetry, $A\,{}_{1}$ corresponds to $\Sigma^{+}/\Delta$.
Thus, the $\,{}^{2}\Delta$ state arose from the calculation of
$\,{}^{2}\Sigma^{+}$. Molecular orbitals for the subsequent CI calculations
were obtained for each spin and symmetry species from state averaged CASSCF
calculations (Werner & Knowles, 1985; Knowles & Werner, 1985) where the state
averaging was achieved over all six electronic states considered in this work.

Within the $C_{2v}$ point group symmetry 14 (8$\sigma$, 3$\pi_{x}$,
3$\pi_{y}$) orbitals which contained 6 closed (4$\sigma$, 1$\pi_{x}$,
1$\pi_{y}$) orbitals were used for all ab initio calculations. Thus, the
active space represents 8 active (4$\sigma$, 2$\pi_{x}$, 2$\pi_{y}$) orbitals
with 9 active electrons and spans the valence orbitals
$5\sigma$–$8\sigma$,$2\pi,3\pi$. All CI calculations were carried out with the
internally contracted CI method (Werner & Knowles, 1988; Knowles & Werner,
1988). The CI calculations used the same set of reference configurations used
in the CASSCF calculation.

The EAMCs represent the $L_{x}$ and $L_{y}$ matrix elements, which are
important for the accurate description of the lambda doubling effects in $\Pi$
states originating from the interactions with the $\Sigma$ and $\Delta$
states. In Fig. 2 (right) the $L_{x}$ components are shown, with $L_{y}$
related to them by symmetry.

The definition of SOCs, which help form a complete and self-consistent
description of a spectroscopic model, requires knowledge of the magnetic
quantum numbers $M_{S}$ (i.e. the projection of the electronic spin $\Sigma$),
which are specified in Table 1.

All of the ab initio curves are later mapped to a different, denser grid used
in the solution of the rovibronic problem via interpolation and extrapolation
as described in the Spectroscopic Model section below.

Figure 1: Ab initio PECs calculated at the icMRCI level of theory using aug-
cc-pVQZ basis set. The grid was specifically made denser around the
equilibrium points of the $X\,{}^{2}\Sigma^{+}$ and $A\,{}^{2}\Pi$ states.

Figure 2: Ab initio coupling curves of SiN at the icMRCI/aug-cc-pVQZ level of
theory. Left: spin-orbit matrix elements $\langle i|$SO$|j\rangle$ for SiN at
the icMRCI level of theory using aug-cc-pVQZ basis set. The MOLPRO values of
the magnetic quantum numbers $M_{S}$ ($\Sigma$) for the curves can be found in
Table. 1. Right: Electronic angular momentum $L_{x}$ curves.

Figure 3: Ab initio transitional and diagonal dipole moment curves calculated at the icMRCI level of theory using aug-cc-pVQZ basis set. Table 1: MOLPRO magnetic quantum numbers $M_{S}$ values (also known as $\Sigma$, projection of $\hat{\bf S}$ along the body fixed $z$ axis) for the ab initio spin-orbit matrix elements displayed in Fig. 2.  SOC | bra $M_{S}$ | ket $M_{S}$
---|---|---
$\langle$ $A\,{}^{2}\Pi$y $\lvert$ | SOx |  $\rvert$$X\,{}^{2}\Sigma^{+}$$\rangle$ | -0.5 | 0.5
$\langle$ $A\,{}^{2}\Pi$y $\lvert$ | SOx |  $\rvert$$B\,{}^{2}\Sigma^{+}$$\rangle$ | -0.5 | 0.5
$\langle$ $A\,{}^{2}\Pi$y $\lvert$ | SOx |  $\rvert$$a\,{}^{4}\Sigma^{+}$$\rangle$ | -0.5 | 0.5
$\langle$ $D\,{}^{2}\Delta$xy $\lvert$ | SOx |  $\rvert$$A\,{}^{2}\Pi$${}_{y}\rangle$ | -0.5 | 0.5
$\langle$ $X\,{}^{2}\Sigma^{+}$$\lvert$ | SOx |  $\rvert$$b\,{}^{4}\Pi$${}_{y}\rangle$ | 0.5 | -0.5
$\langle$ $B\,{}^{2}\Sigma^{+}$$\lvert$ | SOx |  $\rvert$$b\,{}^{4}\Pi$${}_{y}\rangle$ | 0.5 | -0.5
$\langle$ $A\,{}^{2}\Pi$x $\lvert$ | SOz |  $\rvert$$A\,{}^{2}\Pi$${}_{x}\rangle$ | 0.5 | 0.5
$\langle$ $A\,{}^{2}\Pi$x $\lvert$ | SOz |  $\rvert$$b\,{}^{4}\Pi$${}_{x}\rangle$ | 0.5 | 0.5
$\langle$ $b\,{}^{4}\Pi$x $\lvert$ | SOz |  $\rvert$$b\,{}^{4}\Pi$${}_{x}\rangle$ | 0.5 | 0.5

The (transition) DMCs $A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$,
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$,
$B\,{}^{2}\Sigma^{+}$–$A\,{}^{2}\Pi$, $D\,{}^{2}\Delta$–$A\,{}^{2}\Pi$,
$a\,{}^{4}\Sigma^{+}$–$b\,{}^{4}\Pi$, and DMCs
$X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$–$A\,{}^{2}\Pi$,
$B\,{}^{2}\Sigma^{+}$–$B\,{}^{2}\Sigma^{+}$,
$D\,{}^{2}\Delta$–$D\,{}^{2}\Delta$,
$a\,{}^{4}\Sigma^{+}$–$a\,{}^{4}\Sigma^{+}$ and $b\,{}^{4}\Pi$–$b\,{}^{4}\Pi$
were calculated ab initio at the same level of theory as the PECs and are
shown in Fig. 3. The phases of these non-diagonal TDMCs were selected to be
consistent with the phases of the ab initio curves produced in our MOLPRO
calculations (Patrascu et al., 2014). There is a discontinuity in
$\langle$$X\,{}^{2}\Sigma^{+}$$|\mu_{x}|$$B\,{}^{2}\Sigma^{+}$$\rangle$ at 2.5
Å that we attribute to the interaction with the higher electronic states,
which are not included in this model. Due to the large displacement from the
equilibrium corresponding to very high energies this discontinuity does not
provide any material effect on our results, as can be seen from the spectra
reproduced below.

##  3 MARVEL

All available experimental transition frequencies of SiN were extracted from
the published spectroscopic literature and analysed using the MARVEL procedure
(Furtenbacher et al., 2007; Császár et al., 2007; Furtenbacher & Császár,
2012; Tóbiás et al., 2019). This procedure takes a set of assigned transition
frequencies with measurement uncertainties and converts it into a consistent
set of empirical energy levels with the uncertainties propagated from the
input transitions. The transition data extracted as part of this work from the
literature covers the three main bands of SiN involving the
$X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$, and $B\,{}^{2}\Sigma^{+}$ electronic
states: $X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$,
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$and
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$, as summarised in Table 2.

###  3.1 Description of experimental sources

All the available sources of experimental transitions considered in this work
are listed below:

92ElHaGu by Elhanine et al. (1992): an infrared (IR) study of the
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ system through 724 transitions.
Unfortunately, the original line data with assignments were lost/unavailable
and only the derived spectroscopic constants for $A\,{}^{2}\Pi$ and
$X\,{}^{2}\Sigma^{+}$ states remain. Using their $X\,{}^{2}\Sigma^{+}$
constants we produced four $X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ pseudo-
experimental lines (MARVEL Magic numbers) to help connecting spectroscopic
networks in the MARVEL analysis. We also used their extended set of the
spectroscopic constants in PGOPHER (Western, 2017) to generate pseudo-
experimental energies for the first five vibrational states of $A\,{}^{2}\Pi$
and the first three vibrational states of $X\,{}^{2}\Sigma^{+}$ states (up to
$J=49.5$) for the refinement of our spectroscopic model (see below). Elhanine
et al. (1992) is the only existing information on the vibrationally excited
$A\,{}^{2}\Pi$ states of SiN and is therefore crucial for providing MARVEL
energies for states of $A\,{}^{2}\Pi$ with $v>1$. The PGOPHER file used to
generate the energies for fitting is provided as part of the supplementary
information.

85FoLuAm by Foster et al. (1985): This IR observation reported 187 lines of
the $A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ system originally assigned to the
(2,0) band and 6 microwave transitions in the
$X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ system. This band was later
reassigned to 1–0 by Yamada et al. (1988).

85YaHiXX by Yamada & Hirota (1985): This IR study reported 170 lines of the
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ system originally assigned to the (1,0)
band, which however was later reassigned to (0,0) by Yamada et al. (1988).

76BrDuHo by Bredohl et al. (1976): This is a UV study of the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ system. Only spectroscopic
constants were reported, which we used to generate four pseudo-experimental
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ lines using PGOPHER to help
connect the MARVEL inputs into a single spectroscopic network with
$J^{\prime}=1.5$ $B\,{}^{2}\Sigma^{+}$ $\longleftarrow$ $J"=0.5$
$A\,{}^{2}\Pi$ for the (1,0) and (2,0) bands ($e$ and $f$).

75Linton by Linton (1975): This UV study reported 460 lines of the
$C\,{}^{2}\Pi$–$A\,{}^{2}\Pi$ system. Sadly due to the low quality of this
data and the lack of information provided in the paper to reconstruct the full
set of quantum numbers required for MARVEL, this source is omitted from the
MARVEL analysis of the current work, but the original lines scanned using OCR
(optical character reader) software are still provided as part of the
supplementary information.

68NaVe by Nagaraj & Verma (1968): This UV study reported 422 lines of the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ system covering the (0,0), (1,1),
(2,2) bands. Originally the spectra was assigned to the SiO+ molecules, but
that was reassigned to SiN later as per the correction in Dunn et al. (1969).

29JeDe by Jenkins & de Laszlo (1928): This 1929 work is still the most
extensive to date UV observation of the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ system of SiN reporting 1355 lines
from the (0,0), (1,1), (2,2), (3,3), (4,4), (3,2), (4,3), (5,4), (6,5)
vibronic bands.

In total, 1987 experimental and 9 pseudo-experimental transitions were
processed via the online MARVEL app (available through a user-friendly web
interface at http://kkrk.chem.elte.hu/marvelonline) using the Cholesky
(analytic) approach with a 0.5 cm-1 threshold on the uncertainty of the “very
bad” lines. The final MARVEL process for 28Si14N resulted in one main
spectroscopic network, containing 1054 energy levels and 1456 validated
transitions, with the rotational excitation up to $J=44.5$ and covering
energies up to 30 308 cm-1. These energy levels in conjunction with energies
generated from PGOPHER were used to refine our ab initio rovibronic
spectroscopic model (PECs, SOCs and EAMCs) presented above. The MARVEL input
transitions and output energy files are given as part of the supplementary
material.

Table 2: Breakdown of the assigned transitions by electronic bands for the
sources used in this MARVEL study. A and V are the numbers of the available
and validated transitions, respectively. The mean and maximum uncertainties
(Unc.) obtained using the MARVEL procedure are given in cm-1.

Electronic Band | Vibrational Bands |  $J$ Range | A/V | WN range cm-1 | Unc. (Mean/Max)
---|---|---|---|---|---
29JeDeXX |  |  |  |  |
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ | (0,0),(1,1),(2,2),(3,2),(3,3) | 0.5–43.5 | 1060/1355 | 23267–24279 | 0.100/0.197
| (4,3),(4,4),(5,4),(5,5),(6,5) | 0.5–43.5 |  |  |
68NaVexx |  |  |  |  |
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ | (0,0),(1,1),(2,2) | 0.5–44.5 | 420/422.0 | 23876–24280 | 0.030/0.191
76BrDuHo |  |  |  |  |
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ | (1,0),(2,0) | 0.5–1.5 | 4/4 | 25236–26203 | 0.100/0.100
84YaHiXX |  |  |  |  |
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ | (0,0) | 0.5–31.5 | 195/195 | 1918–2037 | 0.007/0.007
85FoLuAm $A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ | (1,0) | 0.5–36.5 | 199/199 | 2917–3039 | 0.001/1.00$\times 10^{-3}$
$X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ | (0,0) | 0.5–4.5 | 6/6 | 2.9039–5.8329 | 1.00$\times 10^{-3}$/1.00$\times 10^{-3}$
92ElHaGu $X\,{}^{2}\Sigma^{+}$-$X\,{}^{2}\Sigma^{+}$ | (1,0),(2,0) | 0.5–1.5 | 4/4 | 1140–2267 | 0.100/0.100

##  4 Rovibronic calculations

To obtain rovibronic energies and wavefunctions for the six electronic states
in question, we solve a set of fully coupled Schrödinger equation for the
motion of nuclei using the Duo program (Yurchenko et al., 2016). Duo uses the
Hunds case a basis set with the vibrational basis functions obtained by
solving uncoupled vibrational Schrödinger equations for each electronic state
in question with using a sinc DVR basis set (Guardiola & Ros, 1982). Atomic
masses are used to represent the kinetic energy operator. In Duo calculations,
an equidistant grid of 501 radial points ranging from 1.1 to 5.0 Å. The ab
initio curves were cubic-spline interpolated to map the ab initio curves onto
the denser Duo grid. For the extension outside the ab initio bond lengths
($r>3.080$ Å), the ab initio curves were extrapolated using the functional
forms given by (Yurchenko et al., 2016)

$\displaystyle f_{\rm PEC}^{\rm short}(r)$ $\displaystyle=$ $\displaystyle
A+B/r,$ $\displaystyle f_{\rm TDMC}^{\rm short}(r)$ $\displaystyle=$
$\displaystyle Ar+Br^{2},$ (1) $\displaystyle f_{\rm other}^{\rm short}(r)$
$\displaystyle=$ $\displaystyle A+Br,$

for short range and

$\displaystyle f_{\rm PEC}^{\rm long}(r)$ $\displaystyle=$ $\displaystyle
A+B/r^{6}$ $\displaystyle f_{\rm EAMC}^{\rm long}(r)$ $\displaystyle=$
$\displaystyle A+Br,$ (2) $\displaystyle f_{\rm other}^{\rm long}(r)$
$\displaystyle=$ $\displaystyle A/r^{2}+B/r^{3}$

for long range, where $A$ and $B$ are stitching parameters. A more detailed
description of the Duo methodology was previously given by Yurchenko et al.
(2016), see also Tennyson et al. (2016b).

###  4.1 Refining the spectroscopic model

For our spectroscopic model of SiN we initially used ab initio PECs for the
doublets $X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$, $B\,{}^{2}\Sigma^{+}$,
$D\,{}^{2}\Delta$ and quartets $b\,{}^{4}\Pi$, $a\,{}^{4}\Sigma^{+}$, as well
as all appropriate SOCs and EAMCs of SiN. The $X\,{}^{2}\Sigma^{+}$,
$A\,{}^{2}\Pi$ and $B\,{}^{2}\Sigma^{+}$PECs and the associated couplings were
then refined by fitting to our empirical set of MARVEL and PGOPHER term values
of 28Si14N as described above.

For the refinements, the PECs for the $X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$,
and $B\,{}^{2}\Sigma^{+}$ states were parameterised using the Extended Morse
Oscillator (EMO) function (Lee et al., 1999) as given by

$V(r)=V_{\rm e}\;\;+\;\;(A_{\rm e}-V_{\rm
e})\left[1\;\;-\;\;\exp\left(-\sum_{k=0}^{N}B_{k}\xi_{p}^{k}(r-r_{\rm
e})\right)\right]^{2},$ (3)

where $A_{\rm e}-V_{\rm e}=D_{\rm e}$ is the dissociation energy, $A_{\rm e}$
is the corresponding asymptote, $r_{\rm e}$ is an equilibrium distance, and
$\xi_{p}$ is the Šurkus variable (Šurkus et al., 1984) given by

$\xi_{p}=\frac{r^{p}-r^{p}_{\rm e}}{r^{p}+r^{p}_{\rm e}}$ (4)

with $V_{\rm e}=0$ for the $X\,{}^{2}\Sigma^{+}$ state. The
$X\,{}^{2}\Sigma^{+}$ and $A\,{}^{2}\Pi$ states have a common asymptote which
was fixed for the analytical form of the potential to the ground state
dissociation energy $D_{\rm e}$ 4.75 eV based on the experiment by Naulin et
al. (1993) (see also Kerkines & Mavridis (2005)). Similarly the
$B\,{}^{2}\Sigma^{+}$ state was adjusted to be 19233 cm-1 above the
$X\,{}^{2}\Sigma^{+}$ dissociation using the atomic excitation energy of N
(Kramida et al., 2021). The processed and refined PECs used in Duo can be seen
in Fig. 4.

Figure 4: Potential energy curves of SiN representing our final spectroscopic
and used in the Duo calculations: $X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$ and
$B\,{}^{2}\Sigma^{+}$ are refined while $D\,{}^{2}\Delta$,
$a\,{}^{4}\Sigma^{+}$ and $b\,{}^{4}\Pi$ are ab initio.

For the parameterisation of SOCs and the EAMCs, the ab initio curves were
morphed using a polynomial decay expansion as given by:

$F(r)=\sum^{N}_{k=0}B_{k}\,z^{k}(1-\xi_{p})+\xi_{p}\,B_{\infty},$ (5)

where $z$ is the damped-coordinate polynomial given by:

$z=(r-r_{\rm ref})\,e^{-\beta_{2}(r-r_{\rm ref})^{2}-\beta_{4}(r-r_{\rm
ref})^{4}}.$ (6)

The refined curves $f(r)$ are the represented as

$f(r)=F(R)f^{\rm ai}(r),$

with $B_{\infty}=1$ in order for $F(r)\to 1$ at $r\to\infty$. Morphing allows
one to retain the original shape of the property with a minimum number of
varied parameters, see e.g. Prajapat et al. (2017) and Yurchenko et al.
(2018). In Eq. (6), $r_{\rm ref}$ is a reference position chosen to be close
to $r_{\rm e}$ of $X\,{}^{2}\Sigma^{+}$ and $\beta_{2}$ and $\beta_{4}$ are
damping factors, typically chosen to be $8\times 10^{-1}$ and $2\times
10^{-2}$. The morphed and extended by Duo SOCs and EAMCs are shown in Fig. 5.

Figure 5: Refined electronic angular momentum and spin-orbit curves for SiN.
The MOLPRO values of the magnetic quantum numbers $M_{S}$ for the SOCs can be
found in Table. 1

The 1062 MARVEL and 854 PGOPHER energy levels values were used to refine the
PECs, SOCs, EAMCs into analytical form as described above. Figure 6 shows the
residuals representing how well our model compares to the MARVEL and PGOPHER
energies. Unfortunately the most significant work from the number of
transitions provided is Jenkins & de Laszlo (1928), where the accuracy of the
transitions was 0.1 cm-1 at best. This is the main reason for the spread of
uncertainties in the $X\,{}^{2}\Sigma^{+}$ and $B\,{}^{2}\Sigma^{+}$ states.
Similarly most data for the $A\,{}^{2}\Pi$ state had to be supplemented using
the PGOPHER calculations due to the difficulty of obtaining experimental data
for the state.

Figure 6: The residuals (Obs.-Calc.) between the experimentally determined
energies of SiN ($X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$ and
$B\,{}^{2}\Sigma^{+}$) from our MARVEL analysis (open), pseudo-experimental
PGOPHER (filled) generated energies and Duo energies corresponding to our
refined spectroscopic model. Only MARVEL energies are available for
$B\,{}^{2}\Sigma^{+}$, hence the vibrational level labels are not
differentiated by source.

###  4.2 Dipole moment curves

Most of the (transition) dipole moment curves from Fig. 3 were left unchanged
and used as grid points in Duo calculations apart from the
$\langle$$X\,{}^{2}\Sigma^{+}$$|\mu_{z}|$$X\,{}^{2}\Sigma^{+}$$\rangle$, which
was fitted to analytical form of the polynomial decay given in Eq. (5). Having
an analytical form helps to reduce the numerical noise arising due to
interpolation of ab initio curves present in high overtone transitions, see
Medvedev et al. (2016). The comparison between ab initio
$X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ DMC and its analytical form is
shown in Fig. 7, with the deviations at very large radial displacement not
affecting the intensities for the energy excitations selected for this study.

Figure 7: Comparison of the ab initio and fitted DMC for the
$X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ system.

All expansion parameters or curves defining our spectroscopic model are given
as supplementary material to the paper as a Duo input file.

##  5 Line list and simulations of spectra of SiN

The SiNful line list was produced with Duo using the empirically refined and
ab initio curves as described above. For the main isotopologue 28Si14N, it
contains 43 646 806 transitions and 131 935 states for $X\,{}^{2}\Sigma^{+}$,
$A\,{}^{2}\Pi$, $B\,{}^{2}\Sigma^{+}$, $D\,{}^{2}\Delta$,
$a\,{}^{4}\Sigma^{+}$and $b\,{}^{4}\Pi$, covering wavenumbers up to 58 000
cm-1 $v=0\ldots 30$ and $J=0\ldots 245.5$. For the isotopologue line lists
only the atomic masses were adjusted in the Duo input files. Further details
on the line list statistics covering the isotpologues can be seen in Table 3.
The line list is provided in State and Transition files, as is customary for
the ExoMol format (Tennyson et al., 2020). Extracts from the States and Trans
files are shown in Tables 4 and 5, respectively; the full files are available
from www.exomol.com. The States file contains energy term values, state
uncertainties, Landé-$g$ factors (Semenov et al., 2017), lifetimes (Tennyson
et al., 2016a) and quantum numbers. The Transition file contains Einstein A
coefficients. The partition functions are also included as part the standard
line list compilation.

Table 3: Line list statistics for each isotopologue of SiN.

Isotopologue | $g_{\rm ns}$ | $N_{\rm states}$ | $N_{\rm trans}$
---|---|---|---
28Si14N | 3 | 131935 | 43646806
29Si14N | 6 | 132335 | 43946969
30Si14N | 3 | 132706 | 44223730
28Si15N | 2 | 133460 | 44816182

$g_{\rm ns}$: Nuclear spin degeneracy;
$N_{\rm states}$: Number of states;
$N_{\rm trans}$: Number of transitions.

The calculated energies were replaced with the MARVEL values (MARVELised),
where available. We have used the labels ’Ca’,’EH’ and ’Ma’ in the penultimate
column of the States file to indicate if the energy value is calculated using
Duo, derived using PGOPHER or MARVELised, respectively.

The uncertainty values in the States file correspond to two cases: the MARVEL
uncertainties are used for MARVELised energies, while for the calculated
values the following approximate expression is used:

${\rm unc}=av+bJ(J+1),$ (7)

where $a$ and $b$ are electronic state dependent constant, given in Table 6.
For the $X\,{}^{2}\Sigma^{+}$, $A\,{}^{2}\Pi$ and $B\,{}^{2}\Sigma^{+}$ states
uncertainties were estimated based on the progression of residuals from Fig. 6
as average increases of obs.-calc. in $v$ and $J$ for each state shown.

Table 4: An extract from the states file of the SiNful line list for 28Si14N.

$i$ |  Energy (cm-1) | $g_{i}$ | $J$ | unc | $\tau$ | $g$ | Parity | State | $v$ | ${\Lambda}$ | ${\Sigma}$ | $\Omega$ | Label | Calc.
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
2 | 1138.471940 | 6 | 0.5 | 0.200000 | 0.5211 | 2.001919 | + | e | X2Sigma+ | 1 | 0 | 0.5 | 0.5 | Ma | 1138.405398
3 | 2017.654928 | 6 | 0.5 | 0.010000 | 0.0015 | 0.000607 | + | e | A2Pi | 0 | 1 | -0.5 | 0.5 | Ma | 2017.651598
4 | 2263.632470 | 6 | 0.5 | 0.400000 | 0.1714 | 2.000967 | + | e | X2Sigma+ | 2 | 0 | 0.5 | 0.5 | Ma | 2263.834585
5 | 3037.121068 | 6 | 0.5 | 0.006000 | 0.0006 | 0.000570 | + | e | A2Pi | 1 | 1 | -0.5 | 0.5 | Ma | 3037.120364
6 | 3375.983139 | 6 | 0.5 | 1.000000 | 0.0368 | 2.001071 | + | e | X2Sigma+ | 3 | 0 | 0.5 | 0.5 | Ma | 3376.117442
7 | 4044.188981 | 6 | 0.5 | 0.200375 | 0.0004 | 0.000245 | + | e | A2Pi | 2 | 1 | -0.5 | 0.5 | EH | 4044.208257
8 | 4475.345990 | 6 | 0.5 | 1.600000 | 0.0124 | 2.001414 | + | e | X2Sigma+ | 4 | 0 | 0.5 | 0.5 | Ma | 4475.285785
9 | 5038.850133 | 6 | 0.5 | 0.300375 | 0.0003 | -0.000102 | + | e | A2Pi | 3 | 1 | -0.5 | 0.5 | EH | 5038.860507
10 | 5561.363517 | 6 | 0.5 | 0.100750 | 0.0057 | 2.001737 | + | e | X2Sigma+ | 5 | 0 | 0.5 | 0.5 | Ca | 5561.363517
11 | 6021.067289 | 6 | 0.5 | 0.400375 | 0.0002 | -0.000370 | + | e | A2Pi | 4 | 1 | -0.5 | 0.5 | EH | 6021.017509
12 | 6634.354317 | 6 | 0.5 | 0.120750 | 0.0032 | 2.001917 | + | e | X2Sigma+ | 6 | 0 | 0.5 | 0.5 | Ca | 6634.354317
13 | 6990.593938 | 6 | 0.5 | 0.500375 | 0.0002 | -0.000488 | + | e | A2Pi | 5 | 1 | -0.5 | 0.5 | Ca | 6990.593938
14 | 7694.239922 | 6 | 0.5 | 0.140750 | 0.0020 | 2.001777 | + | e | X2Sigma+ | 7 | 0 | 0.5 | 0.5 | Ca | 7694.239922
15 | 7947.534233 | 6 | 0.5 | 0.600375 | 0.0002 | -0.000299 | + | e | A2Pi | 6 | 1 | -0.5 | 0.5 | Ca | 7947.534233
16 | 8740.963173 | 6 | 0.5 | 0.160750 | 0.0014 | 2.000443 | + | e | X2Sigma+ | 8 | 0 | 0.5 | 0.5 | Ca | 8740.963173
17 | 8891.787413 | 6 | 0.5 | 0.700375 | 0.0001 | 0.001068 | + | e | A2Pi | 7 | 1 | -0.5 | 0.5 | Ca | 8891.787413

$i$: State counting number.

$\tilde{E}$: State energy term values in cm-1.

$g_{i}$: Total statistical weight, equal to ${g_{\rm ns}(2J+1)}$.

$J$: Total angular momentum.

unc: Uncertainty, cm-1.

$\tau$: Lifetime (s-1).

$g$: Landé $g$-factors.

$+/-$: Total parity.

State: Electronic state.

$v$: State vibrational quantum number.

$\Lambda$: Projection of the electronic angular momentum.

$\Sigma$: Projection of the electronic spin.

$\Omega$: Projection of the total angular momentum, $\Omega=\Lambda+\Sigma$.

Label: ’Ma’ is for MARVEL,’EH’ is PGOPHER generated and ’Ca’ is for Calculated
(using Duo).

Table 5: An extract from the transitions file of the SiNful line list for
28Si14N.

$f$ | $i$ |  $A_{fi}$ (s-1) | $\tilde{\nu}_{fi}$
---|---|---|---
34822 | 33867 | 6.3723E+01 | 4047.041414
2808 | 2455 | 2.0314E-04 | 4047.041433
63044 | 62682 | 1.4457E-06 | 4047.041573
61850 | 60890 | 1.5342E-01 | 4047.042750
65187 | 65424 | 1.1950E+02 | 4047.042900
75707 | 75959 | 2.7837E+00 | 4047.043952
37830 | 36872 | 9.1146E-02 | 4047.044029
49826 | 49468 | 9.3588E-04 | 4047.044408
18065 | 18306 | 6.1554E+01 | 4047.045440
53117 | 53365 | 5.3547E-02 | 4047.046876
70272 | 69910 | 5.1581E-03 | 4047.047010
106232 | 106467 | 5.6296E-01 | 4047.047815

$f$: Upper state counting number;
$i$: Lower state counting number;
$A_{fi}$: Einstein-$A$ coefficient in s-1;
$\tilde{\nu}_{fi}$: transition wavenumber in cm-1.

Table 6: $a$ and $b$ constants (cm-1) defining state dependent uncertainties via Eq. (7). State | $a$ | $b$
---|---|---
$X\,{}^{2}\Sigma^{+}$ | 0.02 | 0.001
$A\,{}^{2}\Pi$ | 0.1 | 0.001
$B\,{}^{2}\Sigma^{+}$ | 0.5 | 0.001
All other | 0.8 | 0.001

###  5.1 Overall Spectra

To demonstrate the accuracy of the SiNful line list, several spectra were
calculated, analysed and compared to available laboratory measurements. Figure
8 illustrates the main bands of SiN at 2000 K. The dominance of the
$X\,{}^{2}\Sigma^{+}$–$A\,{}^{2}\Pi$ in the 0–5000 cm-1 region confirms why it
is hard to detect the comparatively weak
$X\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ transitions. Additionally, while
the $X\,{}^{2}\Sigma^{+}$–$B\,{}^{2}\Sigma^{+}$ band system appears to be
overall stronger than $A\,{}^{2}\Pi$– $B\,{}^{2}\Sigma^{+}$, the latter should
still be detectable in the 18000 – 24000 cm-1 region according to our model,
in line with observations of these vibronic bands by Ojha & Gopal (2006) and
Foster et al. (1985). Additionally, in the region above 28000 cm-1 the
$A\,{}^{2}\Pi$ -$D\,{}^{2}\Delta$ band becomes dominant, which agrees with
previous vibronic observations in this region by Bredohl et al. (1976). Figure
9 shows how the spectrum of 28Si14N changes with increasing temperature.

Figure 8: Simulated absorption spectrum of SiN at 2000 K showing the main
bands of the system. A Gaussian profile of the half width of half maximum
(HWHM) of 1 cm-1 was used.  Figure 9: Temperature dependence of the SiN
spectra using the SiNful line list. A Gaussian profile with HWHM of 1 cm-1 was
used.

###  5.2 $B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ band

$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ is the strongest electronic band
in the system, with the largest number of experimental observations due to it
being the easiest to detect. In Figure 10 we simulate the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ (5,4) and (4,3) band at rotational
temperature of 392 K and 412 K respectively to provide direct comparison with
the experiment. The simulated spectra is also adjusted to be in air rather
than in vacuum to align with the experimental spectra. This is achieved by
using the IAU standard of conversion adopted by Morton (1991):

$\lambda_{\rm air}=\frac{\lambda_{\rm vacuum}}{(1.0+2.735182\times
10^{-4}+\frac{131.4182}{\lambda_{\rm
vac}^{2}}+\frac{2.76249\times^{8}}{\lambda_{\rm vac}^{4}})}$ (8)

where $\lambda_{\rm air}$ and $\lambda_{\rm vac}$ are wavelengths in air and
vacuum respectively. The overall agreement on the rotational structure is
within the uncertainty provided.

Figure 10: Simulated absorption spectra of SiN showing the
$B\,{}^{2}\Sigma^{+}$–$X\,{}^{2}\Sigma^{+}$ (5,4) and (4,3) bands at
temperature of 392 K and 412 K respectively. A comparison to the experimental
spectra from Schofield & Broida (1965) is provided. A Gaussian profile HWHM of
0.2 cm-1 was used.

###  5.3 $A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ band

The $A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ band system was first detected by
Jevons (1913), however because of the vicinity of the $A\,{}^{2}\Pi$ and
$X\,{}^{2}\Sigma^{+}$ states it has proved difficult to study experimentally.
Figure 11 shows the comparison of our simulated spectra of the
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ (1,0) band with the experiment of Foster
et al. (1985). The spectrum is simulated at the effective temperature of 740
K. The position of the bands agrees to 0.05 cm-1 which is within our
calculated uncertainties.

Figure 11: Simulated absorption spectrum of SiN at effective temperature of
740 K showing the $A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ (1,0) band. A Gaussian
profile of HWHM of 0.009 cm-1 was used. A comparison with the experimental
spectra from Foster et al. (1985) is provided, with calculated spectra in red
and experimental in black.

Figure 12 shows a comparison of the simulated spectra (shown as sticks) for
different silicon isotopes of the SiN molecule with the experimental spectra
of Yamada et al. (1988). The overall agreement for 28Si14Nis $\sim$0.016 cm-1,
for 29Si14N is $\sim$0.035 cm-1 and for 30Si14N is $\sim$0.055 cm-1. The
intensities were adjusted based on the natural abundance for each silicon
isotope. These are defined as 92.2%, 4.68% and 3.09% for 28Si14N ,29Si14N,
30Si14N respectively. Additionally, in the same range a Q22(0.5) line of
$A\,{}^{2}\Pi$–$X\,{}^{2}\Sigma^{+}$ (0,0) for 29Si14N should be present, but
it was probably too weak to be observed during the experiment. This line is
indicated with the an asterisk.

Figure 12: Comparison of the simulated stick spectrum and experimental
emission spectrum of the SiN isotpologues (28Si14N, 29Si14N, 30Si14N) at 300
K. Straight lines are experimentally observed line positions and dashed lines
with points represent calculated. The asterisk indicates the Q22(0.5) line for
29Si14N not observed experimentally by (Yamada et al., 1988).

###  5.4 Lifetimes

There are few precise experimental measurements of the lifetimes of different
electronic states of SiN in the literature. Walkup et al. (1984) report the
$B\,{}^{2}\Sigma^{+}$ $v=1$ vibrational state to have a lifetime of 200 ns
$\pm 10$ ns ($T_{\rm rot}$ = 500 K). Our calculated value for this vibronic
state is lower: 130 ns ($J=0.5$), slowly increasing to $150$ ns for $J=100$ ns
and even 200 ns for $J=160.5$ ($v=1$, $B\,{}^{2}\Sigma^{+}$). Additionally
there have been several theoretical works (Xing et al., 2018; Kaur & Baluja,
2015) which showed that for the $A\,{}^{2}\Pi$ state the lifetimes decrease
with the vibrational levels as lower levels are much less energetically
accessible due to the proximity of the $X\,{}^{2}\Sigma^{+}$ state. This can
be seen in more detail in Table 7, where we compare our calculated lifetimes
with those of Xing et al. (2018); the agreement is good.

Table 7: Comparison of lifetimes from our current work (A) and Xing et al. (2018) (B). An experimental lifetime 200 $\pm$ 10 ns was reported by Walkup et al. (1984) for $v^{\prime}=1$ of $B\,{}^{2}\Sigma^{+}$. |  $A\,{}^{2}\Pi$/ $\mu$s |  $B\,{}^{2}\Sigma^{+}$/ ns
---|---|---
$v^{\prime}$ | A | B | A | B
0 | 1460 | 1660 | 124 | 129
1 | 610 | 679 | 130 | 136
2 | 385 | 429 | 137 | 143
3 | 281 | 312 | 143 | 150
4 | 221 | 245 | 150 | 157
5 | 182 | 202 | 156 | 164
6 | 155 | 172 | 163 | 170
7 | 135 | 149 | 169 | 176
8 | 121 | 132 | 175 | 181
9 | 109 | 119 | 181 | 187
10 | 98.3 | 109 | 186 | 192
11 | 90.5 | 99.8 | 192 | 196
12 | 84.0 | 92.6 | 197 | 201
13 | 78.5 | 86.6 | 202 | 205
14 | 73.9 | 81.5 | 207 | 210

###  5.5 Partition function

For the partition function of 28Si14N, we follow the ExoMol and HITRAN
(Gamache et al., 2017) convention and include the full nuclear spin. This
means that our convention is different to that of recently reported by Barklem
& Collet (2016) with whom we compare our results in Fig. 13. In order for
comparison to be in equivalent convention, their partition function was
multiplied by the factor of three as 14N has a nuclear spin degeneracy of 3
and 28S has nuclear spin degeneracy of 0. The differences at higher
temperatures can be attributed to incompleteness of the model used by Barklem
& Collet (2016), which is demonstrated by comparing partition functions by
using only $X\,{}^{2}\Sigma^{+}$ state, $X\,{}^{2}\Sigma^{+}$ and
$A\,{}^{2}\Pi$ states and all states: the partition function of Barklem &
Collet (2016) appears to be based on the $X\,{}^{2}\Sigma^{+}$ state only. As
evident from Fig. 13, the contribution from the $A\,{}^{2}\Pi$ state cannot be
neglected due to its low excitation energy.

Figure 13: Comparison of the partition functions for 28Si14N: values from this
work and the work of Barklem & Collet (2016).

Partition functions in for $T=1-3000$ K in steps of 1 K are available the 4
isotopologue, 28Si14N, 29Si14N, 30Si14N and 28Si15N, are available via the
ExoMol website.

##  6 Conclusion

New IR and UV line lists called SiNful for isotopologues of SiN are presented.
SiNful is available from www.exomol.com (Tennyson et al., 2020) and from
www.zenodo.org (European Organization For Nuclear Research & OpenAIRE, 2013).
As part of the line list construction, a MARVEL analysis for 28Si14N was
performed. All experimental line positions from the literature (to the best of
our knowledge) currently available for the $X$–$X$, $A$–$X$ and $B$–$X$
systems were processed to generate a MARVEL set of empirical energies of SiN.
An accurate spectroscopic model for SiN was built: ab initio (T)DMC and
empirically refined PECs, SOCs, EAMCs, using both previously derived MARVEL
energies as well as PGOPHER generated energies.

The line list was MARVELised, where the theoretical energies are replaced with
the MARVEL values (where available). The line list provides uncertainties of
the rovibronic states in order to help in high-resolution applications.
Comparisons of simulated spectra with experiment for both $A$–$X$ and $B$–$X$
system show close agreement. Additionally lifetimes, Landé-g factors and
partition functions are provided. The importance of inclusion of the energies
of the excited state $A\,{}^{2}\Pi$ when computing the partition function of
SiN is demonstrated.

## Acknowledgements

This work was supported by UK STFC under grant ST/R000476/1 and the European
Research Council (ERC) under the European Union’s Horizon 2020 research and
innovation programme through Advance Grant number 883830.

##  7 Data Availability

The data underlying this article are available in the article and in its
online supplementary material, including (1) the Duo input files for each
isotopologue containing all the potential energy, (transition) dipole moment
and coupling curves of SiN used in this work, (2) the MARVEL data set, (3)
digitalised line data from Linton (1975), (4) extracted ab initio curves as
part of the excel, (5) PGOPHER file used to generate energy levels for two low
lying states, (6) the temperature-dependent partition function of 28Si14N up
to 10000 K. The SiNful line lists for 28Si14N, 29Si14N, 30Si14N and 28Si15N
are available from www.exomol.com.

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# Graceful Labellings of Various Cyclic Snakes

Ahmad H. Alkasasbeh<EMAIL_ADDRESS>Danny Dyer<EMAIL_ADDRESS>
Department of Mathematics and Statistics
Memorial University of Newfoundland
St. John’s, Newfoundland
A1C 5S7 Canada

###### Abstract

In this paper, we present a new sufficiency condition to obtain a graceful
labelling for every $kC_{4n}$ snake and use this condition to label every such
snake for $n=1,2,\ldots,6$. Then, we extend this result to cyclic snakes where
the cycles lengths vary. Also, we obtain new results on the (near) graceful
labelling of cyclic snakes based on cycles of lengths $n=6,10,14$, completely
solving the case $n=6$.

##  1 Introduction

Let $G=(V,E)$ be a graph with $m$ edges. Let $f$ be a labelling defined from
$V(G)$ to $\\{0,1,2,\dots,m\\}$ and let $g$ be the induced edge labelling
defined from $E(G)$ to $\\{1,2,\dots,m\\}$ given by $g(uv)=|f(u)-f(v)|,$ for
all $uv\in E.$ The labelling $f$ is graceful if $f$ is an injective mapping
and $g$ is a bijection. If a graph $G$ has a graceful labelling, then it is
graceful.

Alternatively, let $f$ be defined from $V(G)$ to $\\{0,1,2,\dots,m+1\\}$ and
let $g$ be the induced edge labelling defined from $E(G)$ to $A$, where $A$ is
$\\{1,2,\dots,m-1,m\\}$ or $\\{1,2,\dots,m-1,m+1\\}$ given by
$g(uv)=|f(u)-f(v)|,$ for all $uv\in E.$ Then $f$ is near graceful if $f$ is an
injective mapping and $g$ is a bijection. If a graph $G$ has a near graceful
labelling, then it is near graceful. In this paper, every near graceful
labelling we find will omit the vertex label $m$ and the edge label $m$; that
is the codomain of $f$ will be $\\{1,2,\dots,m-1,m+1\\}$ and the codomain of
$g$ will be $\\{1,2,\dots,m-1,m+1\\}$.

A cyclic snake is a connected graph whose block-cutpoint graph is a path and
each of the blocks is isomorphic to a fixed cycle. We define $kC_{n}$ to be a
cyclic snake with $k$ blocks each of which is $C_{n}$. The string of a
$kC_{n}$ is a sequence of integers
$\left(d_{1},d_{2},d_{3},\ldots,d_{k-2}\right)$ where $d_{i}$ is the distance
between the $i$th and $\left(i+1\right)$th cut vertex, counting cut vertices
from one end of the snake to the other. Certainly, for fixed $n$ and $k$, the
string is uniquely determined by the snake and vice versa. Note that if $k=1$
or $k=2$ the snake $kC_{n}$ has no string. A $kC_{n}$ is linear if all entries
in its string are $\left\lfloor\frac{n}{2}\right\rfloor$, and it is even if
all entries in its string are even numbers.

In [5], Rosa showed that all cycles $C_{n}$ with $n\equiv 0$ or $3\ ($mod$\
4)$ are graceful. Further, he introduced a necessary condition for an Eulerian
graph to be graceful, namely if $G$ is a graceful Eulerian graph with $n$
edges, then $n\equiv 0$ or $3\ ($mod$\ 4)$. A $kC_{n}$ is an Eulerian graph,
and hence graceful only if $kn\equiv 0$ or $3\ ($mod$\ 4)$.

Moulton, in [4], proved that a graceful labelling exists for every $kC_{3}$.
Barrientos, in [1], proved that the cycle $C_{n}$ has a near graceful
labelling if and only if $n\equiv 1$ or $2\ ($mod$\ 4)$. This paper also
showed that a graceful labelling exists for every $kC_{4}$, and for particular
cases of snakes for $C_{6}$, $C_{8}$, and $C_{12}$. A complete survey of graph
labellings is presented in A Dynamic Survey of Graph Labelling [2].

We define a variable snake, $n_{1}C_{m_{1}}n_{2}C_{m_{2}}\ldots
n_{i}C_{m_{i}}$, to be a combination of different $n_{j}C_{m_{j}}$, where
$n_{j}C_{m_{j}}$ is connected with $n_{j+1}C_{m_{j+1}}$ by identifying a
vertex in the last cycle of the $n_{j}C_{m_{j}}$ with a vertex in the first
cycle of the $n_{j+1}C_{m_{j+1}}$ (other than the cut vertex). The string for
a variable snake is similar to the string for a $kC_{n}$.

We will represent all the cycle labellings in this paper as $n$-tuples, with
the overline elements indicating the cut vertices, when necessary.

In Figure 1, we have a near gracefully labelled $5C_{6}$ with string
$\left(3,1,2\right)$. We can represent the labelling of the $5C_{6}$ in Figure
1 in five $6$-tuples as follows: $(20$, 16, 17, 15, 18, $\overline{13})$,
$(\overline{13}$, $21$, $11$, $\overline{22}$, $10$, $19)$, $(\overline{22}$,
$\overline{6}$, 25, 8, 23, $9)$, $(\overline{6}$, 26, 4, 27, $\overline{3}$,
$24)$, $(\overline{3}$, 29, 2, 31, 0, $28)$.

$25$$6$$22$$9$$23$$8$$11$$21$$13$$19$$10$$18$$15$$17$$16$$20$$24$$3$$27$$4$$26$$29$$2$$31$$0$$28$
Figure 1: A near gracefully labelled $5C_{6}$.

In [5], Rosa also introduced an $\alpha$-labelling. An $\alpha$-labelling of a
graph $G$ is a graceful labelling with an extra condition which is there
exists an integer $w$ such that for any edge $uv\in G$, either $f(u)\leq
w<f(v)$ or $f(v)\leq w<f(u)$. Any graph with an $\alpha$-labelling is
necessarily bipartite.

We would also like to introduce an analogue of near graceful labellings. An
$\hat{\alpha}$-labelling of a graph $G$ is a near graceful labelling with an
extra condition which is there exists an integer $w$ such that for any edge
$uv\in G$, either $f(u)\leq w<f(v)$ or $f(v)\leq w<f(u)$. Thus the snake in
Figure 1 has an $\hat{\alpha}$-labelling with $w=16$. In fact, by the nature
of our constructions, the main results of this paper could be given in terms
of $\alpha$\- or $\hat{\alpha}$-labellings. While noting this to be true, we
choose instead to state our results in the more familiar language of graceful
and near graceful labellings.

In Lemmas 1.1 to 1.3, we give some results about (not necessarily graceful)
labellings that will be useful later. In this paper, we adopt the convention
that 0 is a natural number. Then, when we write $\left[x,y\right]$ with
$x,y\in\mathbb{N}$ and $x<y$, we are indicating the set
$\left\\{z\in\mathbb{N}|x\leq z\leq y\right\\}$.

If $G$ is any graph and $f$ is any labelling $G$ then we can relabel $G$ by
adding a constant $c$, $h(v)=f(v)+c$. This technique preserves edge labels.

######  Lemma 1.1.

If $c$ is an arbitrary integer and $f$ is a labelling of a graph $G=(V,E),$
defined by $f:V(G)\rightarrow\left[0,m\right]$ then
$h:V\rightarrow\left[c,c+m\right]$ defined by $h(v)=f(v)+c$, is a labelling
that preserves edge labels.

Proof. If $v_{1}v_{2}\in E$ and $f$ is a labelling then
$f(v_{1}v_{2})=\left|f(v_{1})-f(v_{2})\right|$. By definition,
$h(v_{1}v_{2})=\left|h(v_{1})-h(v_{2})\right|=\left|(f(v_{1})+c)-(f(v_{2})+c)\right|$.
But then $h(v_{1}v_{2})=\left|f(v_{1})-f(v_{2})\right|=f(v_{1}v_{2})$.
Therefore, $h$ preserves edge labels.  $\Box$

######  Lemma 1.2.

If $c$ is an arbitrary integer and $f$ is a labelling of a graph $G=(V,E),$
defined by $f:V(G)\rightarrow\left[0,m\right]$ then
$h:V\rightarrow\left[c-m,c\right]$ defined by $h(v)=c-f(v)$, is a labelling
that preserves edge labels. Further if $f$ is graceful and $c=m$, then
$h(v)=m-f(v)$ is graceful.

Proof. This proof follows the same argument as Lemma 1.1.

If $f$ is graceful and $c=m$, then $h(v)=m-f(v)$ is graceful. Let
$v_{1},v_{2}\in V$ such that $h(v_{1})=h(v_{2}).$ Then $m-f(v_{1})=m-f(v_{2})$
which implies $f(v_{1})=f(v_{2});$ since $f$ is injective, then $v_{1}=v_{2}.$
Therefore, $h$ is injective. Since $h$ is also edge-preserving, it is
graceful. $\Box$

In Lemma 1.3 we present a similar result for near graceful labelling; we omit
$1$ from the range of $f$ and the domain of $h$, because if $f(v)=1$ then
$h(v)=m$, and this would contradict the definition of a near graceful
labelling.

######  Lemma 1.3.

If $f$ is a near graceful labelling of a graph $G=(V,E),$
$f:V(G)\rightarrow\\{0,2,3,\dots,m-1,m+1\\}$ then
$h:V(G)\rightarrow\\{0,2,3,\dots,m-1,m+1\\}$ defined by $h(v)=(m+1)-f(v)$ is a
near graceful labelling.

Proof. This proof follows the same argument as Lemma 1.2. $\Box$

We describe the technique in Lemmas 1.2 and 1.3 as taking the complement of a
(near) graceful labelling.

In [5] Rosa introduced a graceful labelling for $C_{4n}$ with $n\geq 1$.

######  Lemma 1.4.

[5] Let $C_{4n}$ be a cycle with $4n$ edges and vertices $v_{i}$, for $1\leq
i\leq 4n$. Then the following labelling $f$ shows that $C_{4n}$ is graceful:

$f(v_{i})=\left\\{\begin{array}[]{ccc}\frac{i-1}{2}&$if$\ i\ $ is odd$,&\\\
4n+1-\frac{i}{2}&$if$\ i\ $ is even$,&i\ \leq\frac{t}{2},\\\
4n-\frac{i}{2}&$if$\ i\ $ is even$,&i>\frac{t}{2}.\end{array}\right.$ (1)

Barrientos in [1] obtained the following results.

######  Theorem 1.5.

[1] The $kC_{4}$ has a graceful labelling for any string.

######  Theorem 1.6.

[1] The linear $kC_{6}$ is near graceful if $k$ is odd and graceful if $k$ is
even.

######  Theorem 1.7.

[1] The even $kC_{8}$ and $kC_{12}$ are graceful graphs.

######  Theorem 1.8.

[1] The even $kC_{4n}$ with string $\left(d_{1},d_{2},\ldots,d_{k-2}\right)$,
where $d_{i}\in\left\\{2,4\right\\},$ has a graceful labelling.

######  Theorem 1.9.

[1] The even $kC_{4n}$, $4\leq n\leq 5$, with string
$\left(d_{1},d_{2},\ldots,d_{k-2}\right)$, where
$d_{i}\in\left\\{2,4,2n\right\\}$ has a graceful labelling.

In this paper, we introduce a new sufficiency condition to get a graceful
labelling for every $kC_{4n}$. Then, we extend this result to
$n_{1}C_{m_{1}}n_{2}C_{m_{2}}\ldots n_{i}C_{m_{i}}$. Further, we extend the
results in Theorem 1.6 to 1.9 on (near) gracefully labelled $kC_{n}$ where
$n=6,8,12,16,20,24$ for all possible strings. Also, we present new results on
the (near) graceful labelling of $kC_{n}$ where $n=10,14$ and $k>1$.

##  2 Graceful Labelling of $kC_{m}$ for $m\equiv 0\ ($mod$\ 4)$

Since the size of a $kC_{4n}$ is $4kn\equiv 0\ ($mod$\ 4)$, we have the
potential to find a graceful labelling for any $kC_{4n}$. In Theorem 2.1, we
give a new sufficient condition which, when satisfied, shows there is a
graceful labelling of a $kC_{4n}$ for any string.

For fixed even $t$ and an arbitrary fixed positive integer $s$, an
$s,t$-useful cycle with even distance $d$ is a $t$-cycle with vertices
labelled from $\left[0,\frac{t}{2}-1\right]\cup\left[st-\frac{t}{2},st\right]$
and edge labels $\left[st-t+1,st\right]$, with a vertex labelled $0$ and a
vertex labelled $\frac{t}{2}-1$ at distance $d$. Similarly, an $s,t$-odd
useful cycle with odd distance $d$ is a $t$-cycle with vertices labelled from
$\left[0,\frac{t}{2}\right]\cup\left[st-\frac{t}{2}+1,st\right]$ and edge
labels $\left[st-t+1,st\right]$, with a vertex labelled $0$ and vertex
labelled $st-\frac{t}{2}+1$ at distance $d$.

A complete $s,t$-useful cycle set is a set of $s,t$-useful cycles of even and
odd distances, the union of whose distances is
$\left\\{1,2,\ldots,\frac{t}{2}\right\\}$. Let $C_{t}^{d}$ be an element of a
complete $s,t$-useful cycle set such that the distance between the vertices
labelled $0$ and $\frac{t}{2}-1$ is $d$ if $d$ is even, and the distance
between the vertices labelled $0$ and $st-\frac{t}{2}+1$ is $d$ if $d$ is odd.

######  Theorem 2.1.

If there is a complete $s,4n$-useful cycle set with $s\geq 1$, then there
exists a graceful labelling of any $kC_{4n}$.

Proof. To prove this result, we will in fact prove a slightly more complex
result: namely, that given a complete $s,4n$-useful cycle set with $s\geq 1$
and for any $k\geq 1$, then there exists a graceful labelling of any $kC_{4n}$
with $0$ in the last cycle in any position except the cut vertex.

If $0$ is in the last cycle of a $kC_{4n}$, then up to symmetry its position
is uniquely determined by the distance from the last cut vertex. These
distances, $d$, can only be $1,2,\ldots,2n$.

We proceed by induction on $k.$ For $k=1$ use the graceful labelling for
$C_{t}$ in Lemma 1.4, letting $t=4n$. The vertex labels for this graceful
$1C_{4n}$ are a subset of $\left[0,4n\right]$ and the edge labels are exactly
$\left[1,4n\right].$

For $k=2$, we label $2C_{4n}$ while obtaining a vertex with label $0$ at even
distance $d$ from the unique cut vertex as follows. We label one cycle with
the labelling used for $1C_{4n}$, with $2n-1$ added to each vertex, so that
the vertex formerly labelled $0$, and now labelled $2n-1$, is the cut vertex
of $2C_{4n}$. The vertices have been labelled from the set
$\left[2n-1,6n-1\right]$ and, by Lemma 1.1, the edge labels are
$\left[1,4n\right].$ Apply the labelling $C_{4n}^{d}$ to the second cycle
where $d$ is even, with the cut vertex receiving the label $2n-1$. Then this
labelling of $2C_{4n}$ has all vertex labels from $\left[0,8n\right],$ and the
edge labels are exactly $\left[1,8n\right],$ with no repeated vertex or edge
label. That is, it is a graceful labelling of $2C_{4n}$ with the vertex
labelled $0$ at even distance $d$ from the cut vertex.

In the same way we label $2C_{4n}$ while obtaining a vertex with label $0$ at
odd distance $d$ from the unique cut vertex. We label one cycle with the
labelling used for $1C_{4n}$, but replace each vertex label $x$ by $6n+1-x$.
The vertices have been labelled from the set $\left[2n+1,6n+1\right]$ and, by
Lemma 1.1, the edge labels are $\left[1,4n\right].$ Apply the labelling
$C_{4n}^{d}$ to the second cycle where $d$ is odd, with the cut vertex
receiving the label $6n+1$. Then this labelling of $2C_{4n}$ has all vertex
labels from $\left[0,8n\right],$ and the edge labels are exactly
$\left[1,8n\right],$ with no repeated vertex or edge label. That is, it is a
graceful labelling of $2C_{4n}$ with the vertex labelled $0$ at odd distance
$d$ from the cut vertex.

Consider an arbitrary $kC_{4n}$ with $k\geq 3$, with the last entry in its
string $d_{k-2}$. Let $G$ be the $(k-1)C_{4n}$ obtained by deleting a last
cycle from this $kC_{4n}$. By the induction hypothesis, there is a graceful
labelling of $G$ with a $0$ on the vertex distance $d_{k-2}$ from the previous
cut vertex. This labelling has vertex labels that are a subset of
$\left[0,4nk-4n\right]$ and the edge labels are exactly
$\left[1,4kn-4n\right]$.

We label $kC_{4n}$ obtaining a vertex with label $0$ at even distance $d$ from
the unique cut vertex as follows. For the first $k-1$ cycles, use the
labelling of $G$ and add $2n-1$ to each vertex label, so that the final cut
vertex receives label $2n-1$. Thus, the vertices have been labelled from the
set $\left[2n-1,4kn-2n-1\right]$ and by Lemma 1.1, the edge labels are
$\left[1,4kn-4n\right]$. Apply the labelling $C_{4n}^{d}$ to the final cycle,
with the cut vertex receiving label $2n-1$. Then this labelling of $kC_{4n}$
has all vertices labelled from $\left[0,4kn\right],$ and the edge labels are
exactly $\left[1,4kn\right],$ with no repeated vertex or edge label. Thus,
there is a graceful labelling of $kC_{4n}$ with the vertex labelled $0$ at
even distance $d$ from the cut vertex.

We label $kC_{4n}$ obtaining a vertex with label $0$ at odd distance $d$ from
the unique cut vertex as follows. For the first $k-1$ cycles, use the
labelling of $G$ with a $0$ on the vertex distance $d_{k-2}$ from the previous
cut vertex. Then subtract each vertex label from $4kn-2n+1$. Thus, the
vertices have been labelled from the set $\left[2n+1,4kn-2n+1\right]$ and by
Lemma 1.1, the edge labels are $\left[1,4kn-4n\right].$ Apply the labelling
$C_{4n}^{d}$ to the final cycle, with the cut vertex receiving label
$4kn-2n+1$. Then this labelling of $kC_{4n}$ has all vertices labelled from
$\left[0,4kn\right],$ and the edge labels are exactly $\left[1,4kn\right],$
with no repeated vertex or edge label. Thus, there is a graceful labelling of
$kC_{4n}$ with the vertex labelled $0$ at odd distance $d$ from the cut
vertex. $\Box$

In Table 1, we give labellings for $C_{4n}^{2j}$ where $1\leq j\leq n$ and
$n\leq 6$. For each $C_{4n}^{2j}$, we can use Lemma 1.2 with $c=t$ to obtain
$C_{4n}^{2j-1}$. Then, $\left\\{C_{4n}^{m}|1\leq m\leq 2n\right\\}$ is a
complete $s,4n$-useful cycle set. Combining these sets with Theorem 2.1, we
obtain the following corollary.

######  Corollary 2.2.

If $1\leq n\leq 6$ and $k\geq 1$ then every $kC_{4n}$ is graceful.

|  Labelling |
---|---|---
$C_{4}^{2}$ |  $\left(0,t,\overline{1},t-2\right)$ |  [1, 3]
$C_{8}^{2}$ |  $\left(0,t,\overline{3},t-4,2,t-3,1,t-1\right)$ |
$C_{8}^{4}$ |  $\left(0,t,1,t-1,\overline{3},t-4,2,t-3\right)$ |  [1]
$C_{12}^{2}$ |  $\left(0,t,\overline{5},t-6,4,t-5,3,t-4,2,t-2,1,t-1\right)$ |  [1]
$C_{12}^{4}$ |  $\left(0,t,4,t-5,\overline{5},t-6,2,t-4,3,t-2,1,t-1\right)$ |
$C_{12}^{6}$ |  $\left(0,t,1,t-1,2,t-6,\overline{5},t-5,4,t-2,3,t-4\right)$ |  [1]
$C_{16}^{2}$ |  $\left(0,t,\overline{7},t-8,6,t-7,5,t-6,4,t-5,3,t-3,2,t-2,1,t-1\right)$ |
$C_{16}^{4}$ |  $\left(0,t,1,t-1,\overline{7},t-8,6,t-7,5,t-6,4,t-5,2,t-2,3,t-3\right)$ |
$C_{16}^{6}$ |  $\left(0,t,1,t-1,2,t-2,\overline{7},t-8,6,t-7,5,t-6,4,t-3,3,t-5\right)$ |
$C_{16}^{8}$ |  $\left(0,t,1,t-1,2,t-2,4,t-3,\overline{7},t-8,6,t-7,5,t-6,3,t-5\right)$ |  [1]
$C_{20}^{2}$ |  $\left(0,t,\overline{9},t-10,8,t-9,7,t-8,6,t-7,5,t-6,4,t-4,3,t-3,2,t-2,1,t-1\right)$ |
$C_{20}^{4}$ |  $\left(0,t,1,t-1,\overline{9},t-10,8,t-9,7,t-8,6,t-7,5,t-6,3,t-4,4,t-2,2,t-3\right)$ |
$C_{20}^{6}$ |  $\left(0,t,1,t-1,2,t-10,\overline{9},t-9,8,t-8,7,t-7,6,t-3,5,t-6,4,t-2,3,t-4\right)$ |
$C_{20}^{8}$ |  $\left(0,t,1,t-4,3,t-6,8,t-9,\overline{9},t-10,6,t-7,4,t-8,7,t-3,5,t-1,2,t-2\right)$ |
$C_{20}^{10}$ |  $\left(0,t,1,t-1,2,t-2,3,t-3,5,t-4,\overline{9},t-10,8,t-9,7,t-8,6,t-6,4,t-7\right)$ |  [1]
$C_{24}^{2}$ |  $\left(0,t,\overline{11},t-12,10,t-11,9,t-10,8,t-9,7,t-8,6,t-7,5,t-5,4,t-4,3,t-3,2,t-2,1,t-1\right)$ |
$C_{24}^{4}$ |  $\left(0,t,1,t-4,\overline{11},t-12,10,t-11,9,t-10,8,t-9,7,t-7,6,t-3,5,t-5,2,t-1,3,t-8,4,t-2\right)$ |
$C_{24}^{6}$ |  $\left(0,t,1,t-1,2,t-12,\overline{11},t-11,10,t-10,9,t-9,8,t-8,7,t-5,6,t-7,3,t-2,4,t-3,5,t-4\right)$ |
$C_{24}^{8}$ |  $\left(0,t,1,t-4,2,t-1,3,t-8,\overline{11},t-12,10,t-11,9,t-7,8,t-10,7,t-5,5,t-9,4,t-3,6,t-2\right)$ |
$C_{24}^{10}$ |  $\left(0,t,1,t-4,2,t-1,3,t-8,4,t-5,\overline{11},t-12,10,t-11,9,t-10,8,t-9,6,t-7,7,t-3,5,t-2\right)$ |
$C_{24}^{12}$ |  $\left(0,t,1,t-1,2,t-2,3,t-3,4,t-5,6,t-4,\overline{11},t-12,10,t-11,9,t-10,8,t-9,7,t-7,5,t-8\right)$ |
Table 1: Useful labellings $C_{4n}^{2j}$, where $t=4kn$.

By adding $t-4n$ to all even vertices in Equation (1) in Lemma 1.4 we obtain
another labelling.

$g(v_{i})=\left\\{\begin{array}[]{ccc}\frac{i-1}{2}&$if$\ i\ $is odd$,&\\\
t+1-\frac{i}{2}&$if$\ i\ $is even$,&i\leq 2n,\\\ \hfill t-\frac{i}{2}&$if$\ i\
$is even$,&i>2n.\end{array}\right.$ (2)

From Equation (2) we obtain $C_{4n}^{2}$ and $n\geq 1$, since the distance
between the labels $0$ and $2n-1$ in $C_{4n}$ is $2$. The same labelling also
works as $C_{4n}^{3}$, since the distance between the labels $0$ and
$t-\left(2n-1\right)$ is $3$. Further, applying Lemma 1.2 with $c=t$ to the
labelling $C_{4n}^{2}$ gives a new labelling with distance $4$ between the
labels $0$ and $2n-1$, $C_{4n}^{4}$. Then, $S=\left\\{C_{4n}^{m}|2\leq m\leq
4\right\\}$ is a $s,4n$-useful cycle set (though not complete). Combining the
set $S$ with Theorem 2.1, we obtain the following theorem.

######  Theorem 2.3.

The snake $kC_{4n}$ with string $\left(d_{1},d_{2},\ldots,d_{k-2}\right)$ has
a graceful labelling if $d_{i}\in\left\\{2,3,4\right\\}$ for all $i$.

From the previous discussion we can gracefully label a variable snake made
from any $kC_{4n}$ with $n\leq 6$. As an example of a variable snake, consider
$3C_{8}$, gracefully labelled via Theorem 2.1, with the vertices labelled from
the set $\left[0,24\right]$ and the edge labels $\left[1,24\right]$. Then form
a new labelling via Lemma 1.1 with $c=5$, so that a vertex in the last cycle
obtains the label $5$. Then add any $C_{12}^{2j}$ (from Table 1) to $3C_{8}$.
We obtain a gracefully labelled $3C_{8}1C_{12}$. More generally, if we have a
complete $k,4i$-useful cycle set for all $1\leq i\leq n$, then, by combining
these sets with Theorem 2.1, we obtain the following corollary.

######  Corollary 2.4.

If there is a complete $s,4i$-useful cycle set with $s\geq 1$ for all $1\leq
i\leq n$, $j\geq 1$, and $1\leq m_{1},m_{2},\ldots,m_{j}\leq n$, with
$n_{1},n_{2},\ldots,n_{j}$ positive integers then every
$n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{j}C_{4m_{j}}$ is graceful.

Proof. We follow the same method as in the proof of Theorem 2.1. That is, we
will proceed by induction on the number of cycles
$k=n_{1}+n_{2}+\cdots+n_{j}$, and will prove the slightly more complex result
that if there is a complete $s,4i$-useful cycle set with $s\geq 1$ for all
$1\leq i\leq n$, $j\geq 1$, and $1\leq m_{1},m_{2},\ldots,m_{j}\leq n$, with
positive integers $n_{1},n_{2},\ldots,n_{j}$ then every
$n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{j}C_{4m_{j}}$ is graceful with the
label $0$ in any position in the last cycle except the cut vertex. For $k=1$
we have a $k,4i$-useful cycle set from Theorem 2.1, so there exists a graceful
labelling of any $1C_{4n}$ with $0$ in any position in the last cycle.

Consider an arbitrary $n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{j}C_{4m_{j}}$
with $k>1$ and $m=\sum\limits_{l=1}^{j}n_{l}\left(4m_{l}\right)$, the total
number of edges. Let $G$ be the graph obtained by deleting a last $4m_{j}$
cycle from $n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{j}C_{4m_{j}}$. By the
induction hypothesis, there is a graceful labelling of $G$ with a $0$ on the
vertex distance $d_{k-2}$ from the previous cut vertex. This labelling has
vertex labels that are a subset of $\left[0,m-4m_{j}\right]$ and the edge
labels are exactly $\left[1,m-4m_{j}\right]$.

We label $n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{j}C_{4m_{j}}$ obtaining a
vertex with label $0$ at even distance $d$ from the last cut vertex as
follows. For the first $k-1$ cycles, use the labelling of $G$ and add
$2m_{j}-1$ to each vertex label, so that the final cut vertex receives label
$2m_{j}-1$. Thus, the vertices have been labelled from the set
$\left[2m_{j}-1,m+2m_{j}-1\right]$ and by Lemma 1.1, the edge labels are
$\left[1,m-4m_{j}\right]$.

Now the labelling of $G$ has the label $2m_{j}-1$ at the last cycle. So apply
the labelling of $C_{4m_{j}}^{d}$ to the last cycle of $G$, with the cut
vertex receiving label $2m_{j}-1$ and $0$ at position $d$ (even distance) from
the cut vertex. Then this labelling has all vertices labelled from
$\left[0,m\right],$ and edge labels exactly $\left[1,m\right]$, with no
repeated vertex or edge label. Thus, there is a graceful labelling of
$n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{i}C_{4m_{j}}$ with the vertex
labelled $0$ at even distance $d$ from the cut vertex.

We label $n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots n_{j}C_{4m_{j}}$ to obtain a
vertex with label $0$ at odd distance $d$ from the last cut vertex as follows.
For the first $k-1$ cycles, use the labelling of $G$ and subtract each vertex
label from $m-2m_{j}+1$, so that the final cut vertex receives label
$m-2m_{j}+1$. Thus, the vertices have been labelled from the set
$\left[2m_{j}-1,m-2m_{j}+1\right]$ and by Lemma 1.1, the edge labels are
$\left[1,m-2m_{j}+1\right]$.

Now the labelling of $G$ has the label $m-2m_{j}+1$ at the last cycle. So
apply the labelling of $C_{4m_{j}}^{d}$ to the last cycle of $G$, with the cut
vertex receiving label $m-2m_{j}+1$ and $0$ at position $d$ (odd distance)
from the cut vertex. Then this labelling has all vertices labelled from
$\left[0,m\right],$ and edge labels exactly $\left[1,m\right]$, with no
repeated vertex or edge label. Thus, there is a graceful labelling of
$n_{1}C_{4m_{1}}n_{2}C_{4m_{2}}\ldots,n_{i}C_{4m_{j}}$ with the vertex
labelled $0$ at odd distance $d$ from the cut vertex. $\Box$

##  3 Graceful Labelling of $kC_{m}$ for $m\equiv 2\ ($mod$\ 4)$

In Section 2, we proved that if there is a complete $s,4n$-useful cycle set
with $s\geq 1$, then there exists a graceful labelling of any $kC_{4n}$ for
$k\geq 1$. The next natural question is can we prove the same results for
$kC_{4n+2}$?

In $kC_{4n}$ we have the nice property that $kC_{4n}$ is always graceful
regardless of the parity of $k$. For $kC_{4n+2}$ we will obtain graceful or
near graceful labellings depending on the value of $k$, because the size of a
$kC_{4n+2}$ is $4kn+2k$ which is congruent to $0$ modulus $4$ for $k$ even and
$4kn+2k$ which is congruent to $2$ modulus $4$ for $k$ odd. Thus, we would
essentially need to find two complete useful cycle sets for $4n+2$ because we
are trying to change a graceful labelling to a near graceful one, or the
reverse.

For $kC_{4n}$ we obtained the complete cycle set by taking the complement as
in Lemmas 1.2 to 1.3. Here for $kC_{4n+2}$ we need to omit $1$ from the
labelling of the the near graceful cycles, because if we use $1$ and take the
complement we will obtain a labelling that is not near graceful. For example,
if we take $\left(0,7,3,1,2,5\right)$ as a labelling of $1C_{6}$, the
complement would have $6$ in the resulting vertex labelling and hence would
not be near graceful. In this section we prove that there exists a (near)
graceful labelling of any $kC_{6}$ (Theorem 3.1), because we found an analogue
of a complete $k,6$-useful cycle set in Table 2, and did not use $1$ for the
near graceful useful cycles. An exhaustive analysis shows that no labelling of
$C_{10}$ exists that uses the labels $\left[0,5\right]$ and
$\left[t-5,t+1\right]\setminus\left\\{t\right\\}$ that omits the label $1$.
Thus, despite their effectiveness in the $4n$-cycle case, complete cycle sets
cannot help us label even $kC_{10}$.

In [1], Barrientos proved that the $kC_{4}$ has a graceful labelling with any
string, as summarized in Theorem 1.5. Recall the result of Barrientos from
Theorem 1.6: that the linear $kC_{6}$ is near graceful if $k$ is odd and
graceful if $k$ is even. In Theorem 3.1 we prove (near) graceful labellings
exist for any $kC_{6}$.

In Table 2 we see four labellings of $C_{6}.$ The labellings $C_{6}^{a}$ and
$C_{6}^{b}$ use edge labels
$\left[6k-5,6k+1\right]\setminus\left\\{6k\right\\}$. The labellings
$C_{6}^{c}$ and $C_{6}^{d}$ use edge labels
$\left[6k-6,6k\right]\setminus\left\\{6k-5\right\\}$.

| Labelling
---|---
$C_{6}^{a}$ |  $\left(0,6k+1,2,6k-1,\overline{3},6k-2\right)$
$C_{6}^{b}$ |  $\left(0,6k+1,\overline{3},6k-2,2,6k-1\right)$
$C_{6}^{c}$ |  $\left(0,6k,\overline{3},6k-3,1,6k-1\right)$
$C_{6}^{d}$ |  $\left(0,6k,1,6k-1,\overline{3},6k-3\right)$
Table 2: Some useful labellings of $C_{6}$.

######  Theorem 3.1.

If $k\geq 1$ then there exists a (near) graceful labelling of any $kC_{6}$.

Proof. As in proof of Theorem 2.1, we prove a slightly more complex result.
Namely, we prove that if $k\geq 1$, then there exists a (near) graceful
labelling of any $kC_{6}$ with $0$ in the last cycle in any position except
the cut vertex.

If $0$ is in the last cycle of a $kC_{6},$ then up to symmetry its position is
uniquely determined by the distance from the last cut vertex. These distances,
$d$, can only be $1,2$ or $3.$

We proceed by induction on $k.$ For $k=1,$ use $C_{6}^{a}$ or $C_{6}^{b}$ in
Table 2 with $k=1$ which will make it near graceful. For $k=2,$ use
$\left(4,7,2,\overline{9},5,6\right)$ and
$\left(\overline{9},0,12,1,11,3\right)$ to obtain a labelling with $d=1$;
$\left(8,5,10,\overline{3},7,6\right)$ and
$\left(\overline{3},12,0,11,1,9\right)$ to obtain a labelling with $d=2$; and
$\left(4,7,2,\overline{9},5,6\right)$ and $\left(9,1,11,0,12,3\right)$ to
obtain a labelling with $d=3$.

Case 1: Consider an arbitrary $kC_{6}$ with $k\geq 4$ and $k$ even, with the
last entry in the string $d_{k-2}$. Let $G$ be the $(k-1)C_{6}$ obtained by
deleting a last cycle from this $kC_{6}$. By the induction hypothesis, there
is a near graceful labelling of $G$ with a $0$ on the vertex distance
$d_{k-2}$ from the previous cut vertex. This labelling has vertex labels that
are a subset of $\left[0,6k-5\right]\setminus\left\\{6k-6\right\\}$ and the
edge labels are exactly $\left[1,6k-5\right]\setminus\left\\{6k-6\right\\}$.

We label $kC_{6}$ obtaining a vertex with label $0$ at distance $d$ from the
cut vertex as follows. For the first $k-1$ cycles, use the labelling of $G$
and add $3$ to each vertex, so that the final cut vertex receives label $3$.
Thus, the vertices have been labelled from the set
$\left[3,6k-2\right]\setminus\left\\{6k-3\right\\}$ and by Lemma 1.1, the edge
labels are $\left[1,6k-5\right]\setminus\left\\{6k-6\right\\}.$

Apply the labelling $C_{6}^{c}$ or $C_{6}^{d}$ to the final cycle, with the
cut vertex receiving label $3$. Then this labelling of $kC_{6}$ has all
vertices labelled from $\left[0,6k\right],$ and the edge labels are exactly
$\left[1,6k\right],$ with no repeated vertex or edge label. Then there is a
graceful labelling of $kC_{6}$ with $0$ in the $d=2$ position.

We obtain a labelling with $0$ in the $d=1$ position by using the previously
discussed labelling, ending with the $C_{6}^{c}$-labelling in the last cycle,
then applying Lemma 1.2. We obtain $d=3$ by using the $C_{6}^{d}$-labelling in
the last cycle, and applying Lemma 1.2.

Case 2: Consider an arbitrary $kC_{6}$ with $k\geq 3$ and $k$ odd, with the
last entry in the string $d_{k-2}$. We proceed in the same fashion as in Case
1, labelling all vertices except those in the final cycle, with vertices
labelled from the set $\left[3,6k-3\right]$ and edges labelled
$\left[1,6k-6\right]$.

Apply the labelling $C_{6}^{a}$ or $C_{6}^{b}$ to the final cycle, with the
cut vertex receiving label $3$. Then this labelling of $kC_{6}$ has all
vertices labelled from $\left[0,6k+1\right]\setminus\left\\{1,6k\right\\}$,
and the edge labels are exactly
$\left[1,6k+1\right]\setminus\left\\{6k\right\\}$, with no repeated vertex or
edge label. Then there is a graceful labelling of $kC_{6}$ with $0$ in the
$d=2$ position.

We obtain a labelling with $0$ in the $d=1$ position by using the previously
discussed labelling, ending with the $C_{6}^{b}$-labelling in the last cycle,
then applying Lemma 1.3. We obtain $d=3$ by using the $C_{6}^{a}$-labelling in
the last cycle, and applying Lemma 1.3.  $\Box$

In Theorem 3.2 we prove (near) graceful labellings exist for some $kC_{10}$.
In Table 3 we see $7$ labellings of $C_{10}.$ The labellings of
$C_{10}^{a},C_{10}^{e},C_{10}^{g}$, and $C_{10}^{h}$ use edge labels
$\left[10k-9,10k+1\right]\setminus\left\\{10k\right\\}$. The labellings of
$C_{10}^{b},C_{10}^{c},C_{10}^{d}$, and $C_{10}^{f}$ use edge labels
$\left[10k-10,10k\right]\setminus\left\\{10k-9\right\\}$. Since we cannot rely
on the uniformity of a complete cycle set, this theorem uses a variety of
different techniques to achieve similar effects.

|  Labelling
---|---
$C_{10}^{a}$ |  $\left(0,10k+1,4,10k-2,3,10k-4,\overline{5},10k-3,1,10k-1\right)$
$C_{10}^{b}$ |  $\left(0,10k,1,10k-7,3,10k-3,\overline{4},10k-1,2,10k-2\right)$
$C_{10}^{c}$ |  $\left(0,10k,2,10k-4,\overline{4},10k-6,1,10k-2,3,10k-1\right)$
$C_{10}^{d}$ |  $\left(0,10k+1,3,10k-2,1,10k-3,4,10k-4,\overline{5},10k-1\right)$
$C_{10}^{e}$ |  $\left(0,10k,2,10k-2,1,10k-7,3,10k-3,\overline{4},10k-1\right)$
$C_{10}^{f}$ |  $\left(0,10k+1,\overline{5},10k-4,4,10k-3,3,10k-2,1,10k-1\right)$
$C_{10}^{g}$ |  $\left(0,10k,\overline{4},10k-6,2,10k-4,3,10k-2,1,10k-1\right)$
Table 3: Some useful labellings of $C_{10}$.

######  Theorem 3.2.

The $kC_{10}$ ($k\geq 1$) with string ($d_{1},d_{2},\ldots,d_{k-2}$) is
graceful if $k$ is even and near graceful if $k$ is odd and one of the
following is true:

  1. 1.

$d_{i}\in\left\\{4,5\right\\}$ if $i$ is odd and $d_{i}=5$ if $i$ is even,
where $1\leq i\leq k-2$,

  2. 2.

$d_{i}\in\left\\{3,4\right\\}$ if $i$ is odd and $d_{i}=4$ if $i$ is even,
where $1\leq i\leq k-2$,

  3. 3.

$d_{i}\in\left\\{2,3\right\\}$ if $i$ is odd and $d_{i}=3$ if $i$ is even,
where $1\leq i\leq k-2$, or

  4. 4.

$d_{i}\in\left\\{1,2\right\\}$ if $i$ is odd and $d_{i}=2$ if $i$ is even,
where $1\leq i\leq k-2$.

Proof. The proof is similar to the proof of Theorem 3.1 with some changes to
the relabelling technique we use on vertex and edge labels.

Case 1: $d_{i}\in\left\\{4,5\right\\}$ if $i$ is odd and $d_{i}=5$ if $i$
even.

As in the proof of Theorem 3.1, we prove that the $kC_{10}$ ($k\geq 1$) with
string ($d_{1},d_{2},\ldots,d_{k-2}$) satisfying the condition of part one
with $0$ in the $d=4$ or $d=5$ position in the last cycle and near graceful if
$k$ is odd with $10k+1$ in the $d=5$ position in the last cycle.

We proceed by induction on $k.$ For $k=1,$ use the labelling of $C_{10}^{a}$
in Table 3 with $k=1$ which will make it near graceful. For $k=2,$ use
$\left(11,8,13,9,\overline{16},5,14,6,12,10\right)$ and
$\left(0,20,2,18,1,\overline{16},3,17,7,19\right)$ to get $d=5$, and take this
labelling with Lemma 1.2 to get $d=4$.

Case 1a: The proof is similar to the proof of Case 1 in the proof of Theorem
3.1. Consider $kC_{10}$ to be an arbitrary snake with a string as indicated in
the condition of Case 1 with $k\geq 4$ and $k$ even. The labelling of
$(k-1)C_{10}$ by the induction hypothesis has vertex labels that are a subset
of $\left[0,10k-9\right]\setminus\left\\{2,10k-10\right\\}$ and the edge
labels are exactly $\left[1,10k-9\right]\setminus\left\\{10k-10\right\\}$ with
$10k-9$ in the vertex distance $d_{k-2}$ from the previous cut vertex.

We label $kC_{10}$ as follows. For the first $k-1$ cycles, use the labelling
of $\left(k-1\right)C_{10}$ obtained by induction and then subtract each
vertex label from $10k-5$. Thus, the vertices have been labelled from the set
$\left[4,10k-5\right]\setminus\left\\{5,10k-7\right\\}$ and by Lemma 1.2, the
edge labels are $\left[1,10k-9\right]\setminus\left\\{10k-10\right\\}$.

Apply the labelling $C_{10}^{b}$ to the final cycle, with the cut vertex
receiving label $4$. Then this labelling of $kC_{10}$ has all vertices
labelled from $\left[0,10k\right],$ and the edge labels are exactly
$\left[1,10k\right],$ with no repeated vertex or edge label. By induction, a
graceful labelling of $kC_{10}$ exists, with $0$ in the $d=4$ position. (Note
that a possible conflict occurs as $\left(10k-5\right)-2=10k-7$, however, in
the labelling of the $(k-1)C_{10}$ no vertex is labelled $2$, therefore we can
use $C_{10}^{b}$ without any restriction).

We obtain a labelling with $0$ in the $d=5$ position by using the previously
discussed labelling, ending with the $C_{10}^{b}$-labelling in the last cycle,
then applying Lemma 1.2.

Case 1b: Consider $kC_{10}$ to be an arbitrary snake with a string as
indicated in the condition of Case 1 with $k\geq 3$ and $k$ odd. The proof is
similar to the proof of Case 1a, but instead of subtracting each vertex label
from $10k-5$, add $5$ to each vertex and applying the labelling $C_{10}^{a}$
to the final cycle, with the cut vertex receiving label $5$. Then this
labelling of $kC_{10}$ has all vertices labelled from
$\left[0,10k\right]\setminus\left\\{2\right\\}$ and the edge labels are
exactly $\left[1,10k+1\right]\setminus\left\\{10k\right\\}$, with no repeated
vertex or edge label. Then we obtain a near graceful labelling of $kC_{10}$
with $10k+1$ in the $d=5$ position from the cut vertex.

Case 2: $d_{i}\in\left\\{3,4\right\\}$ if $i$ is odd and $d_{i}=4$ if $i$ is
even.

As in the proof of Case 1, we prove that $kC_{10}$ ($k\geq 1$) with string
($d_{1},d_{2},\ldots,d_{k-2}$) satisfying the condition of part two with $0$
in the $d=3$ or $d=4$ position in the last cycle and near graceful if $k$ is
odd with $0$ in the $d=4$ position in the last cycle.

We proceed by induction on $k.$ For $k=1,$ use the labelling of $C_{10}^{a}$
in Table 3 with $k=1$ which will make it near graceful. For $k=2,$ use
$\left(10,9,11,5,13,\overline{4},15,8,12,7\right)$ and
$\left(0,20,2,16,\overline{4},14,1,18,3,19\right)$ to get $d=4$, and take this
labelling with Lemma 1.2 to get $d=3$.

Case 2a: Consider $kC_{10}$ to be an arbitrary snake with a string as
indicated in the condition of Case 2 with $k\geq 4$ and $k$ even. The proof is
similar to the proof of Case 1a.

We label $kC_{10}$ as follows. For the first $k-1$ cycles, use the labelling
of $\left(k-1\right)C_{10}$ obtained by induction and then add $4$ to each
vertex label. Thus, the vertices have been labelled from the set
$\left[4,10k-5\right]\setminus\left\\{6,10k-6\right\\}$ and by Lemma 1.1, the
edge labels are $\left[1,10k-9\right]\setminus\left\\{10k-10\right\\}$.

Apply the labelling $C_{10}^{c}$ to the final cycle, with the cut vertex
receiving label $4$. Then there is a graceful labelling of $kC_{10}$ and $0$
in the $d=4$ position, from the cut vertex. (Note that a possible conflict
occurs as $\left(10k-6\right)+4=10k-2$, however, in the labelling of the
$(k-1)C_{10}$ no vertex is labelled $10k-2$, therefore we can use $C_{10}^{c}$
without any restriction).

We obtain a labelling with $0$ in the $d=3$ position by using the previously
discussed labelling, ending with the $C_{10}^{c}$-labelling in the last cycle,
then applying Lemma 1.2.

Case 2b: Consider $kC_{10}$ to be an arbitrary snake with a string as
indicated in the condition of Case 2 with $k\geq 3$ and $k$ odd. The proof is
similar to the proof of Case 1b, using the labelling of
$\left(k-1\right)C_{10}$ and adding $5$ to each vertex. Finally, apply the
labelling $C_{10}^{a}$ to the final cycle, with the cut vertex receiving label
$5$. Then we obtain a near graceful labelling of $kC_{10}$ and $0$ in the
$d=4$ position, relative to cut vertex.

Case 3: $d_{i}\in\left\\{2,3\right\\}$ if $i$ is odd and $d_{i}=3$ if $i$ is
even.

We prove that the $kC_{10}$ ($k\geq 1$) with string
($d_{1},d_{2},\ldots,d_{k-2}$) satisfying the condition of part three with $0$
in the $d=2$ or $d=3$ position in the last cycle and near graceful if $k$ is
odd with $10k+1$ in the $d=3$ position in the last cycle.

For $k=1,$ use the labelling of $C_{10}^{d}$ in Table 3 with $k=1$ which will
make it near graceful. For $k=2,$ use
$\left(12,6,13,8,\overline{16},5,14,10,11,9\right)$ with $(0$, $20$, $1$,
$\overline{16}$, $3$, $17$, $7$, $19$, $2$, $18)$ to get $d=3$, and take this
labelling with Lemma 1.2 to get $d=2$. The rest of the proof is similar to the
proof of Case 1, but uses $C_{10}^{d}$ instead of $C_{10}^{a}$, and
$C_{10}^{e}$ instead of $C_{10}^{b}$.

Case 4: $d_{i}\in\left\\{1,2\right\\}$ if $i$ is odd and $d_{i}=2$ if $i$ is
even.

We prove that $kC_{10}$ ($k\geq 1$) with string ($d_{1},d_{2},\ldots,d_{k-2}$)
satisfying the condition of part four with $0$ in the $d=1$ or $d=2$ position
in the last cycle and near graceful if $k$ is odd with $10k+1$ in the $d=1$
position in the last cycle.

For $k=1,$ use the labelling of $C_{10}^{f}$ in Table 3 with $k=1$ which will
make it near graceful. For $k=2,$ use
$\left(8,14,7,12,\overline{4},15,6,10,9,11\right)$ with $(0$, $20$, $2$, $18$,
$1$, $13$, $3$, $17$, $\overline{4}$, $19)$ to get $d=2$, and take this
labelling with Lemma 1.2 to get $d=1$. The rest of the proof is similar to the
proof of Case 2, but uses $C_{10}^{f}$ instead of $C_{10}^{a}$, and
$C_{10}^{g}$ instead of $C_{10}^{c}$. $\Box$

In Theorem 3.3 we prove (near) graceful labellings exist for a $kC_{14}$ for
particular strings. In Table 4 we see two labellings of $C_{14}.$ The
labelling $C_{14}^{a}$ uses edge labels
$\left[14k-13,14k+1\right]\setminus\left\\{14k\right\\}$. The labelling
$C_{14}^{b}$ uses edge labels
$\left[14k-14,14k\right]\setminus\left\\{14k-13\right\\}$.

| Labelling
---|---
$C_{14}^{a}$ |  $\left(0,14k+1,4,14k-2,5,14k-4,6,14k-5,\overline{7},14k-6,2,14k-3,1,14k-1\right)$
$C_{14}^{b}$ |  $\left(0,14k,4,14k-10,2,14k-3,3,14k-5,\overline{6},14k-4,5,14k-2,1,14k-1\right)$
Table 4: Some useful labellings of $C_{14}$.

######  Theorem 3.3.

If $k\geq 1$ and $d_{i}\in\left\\{6,7\right\\}$ if $i$ is odd and $d_{i}=7$ if
$i$ is even, for $1\leq i\leq k-2$, then $kC_{14}$ with string
($d_{1},d_{2},\ldots,d_{k-2}$) is graceful if $k$ is even and near graceful if
$k$ is odd.

Proof. The proof is similar to the proof of Theorem 3.1.

As in the proof of Theorem 3.1, we prove that if $k\geq 1$ and
$d_{i}\in\left\\{6,7\right\\}$ if $i$ is odd and $d_{i}=7$ if $i$ is even, for
$1\leq i\leq k-2$, then the $kC_{14}$ with string
($d_{1},d_{2},\ldots,d_{k-2}$) is graceful if $k$ is even with $0$ in the
$d=6$ or $d=7$ position in the last cycle and near graceful if $k$ is odd,
with $14k+1$ in the $d=7$ position in the last cycle.

We proceed by induction on $k.$ For $k=1,$ use the labelling of $C_{14}^{a}$
in Table 4 which will make it near graceful. For $k=2$ use $(21$, $8$, $20$,
$10$, $19$, $13$, $14$, $12$, $15$, $11$, $16$, $9$, $17$, $\overline{6})$ and
$(0$, $28$, $4$, $18$, $2$, $25$, $3$, $23$, $\overline{6}$, $24$, $5$, $26$,
$1$, $27)$ to get $d=6$, and take this labelling with Lemma 1.2 to get $d=7$.

Case 1: Consider $kC_{14}$ to be an arbitrary snake with $0$ in the $d=6$ or
$d=7$ position in the last cycle with $k\geq 4$ and $k$ even. The labelling of
$(k-1)C_{14}$ by the induction hypothesis has vertex labels that are a subset
of $\left[0,14k-15\right]\setminus\left\\{3,14k-14\right\\}$ and the edge
labels are exactly $\left[1,14k+1\right]\setminus\left\\{14k\right\\}$ with
$14k-13$ in the vertex distance $d_{k-2}$ from the previous cut vertex.

We label $kC_{14}$ as follows. Subtract each vertex label from $14k-7$ for the
first $k-1$ cycles of the $(k-1)C_{14}$, so that the last cut vertex receives
label $6$. Then, apply the labelling $C_{14}^{b}$ to the final cycle, with the
cut vertex receiving label $6$. Then this labelling of $kC_{14}$ has all
vertices labelled from $\left[0,14k\right]$ and the edge labels are exactly
$\left[1,14k\right]$, with no repeated vertex or edge labels. Hence, we obtain
a graceful labelling of $kC_{14}$ and $0$ in the $d=6$ position.

By using the previously discussed labelling, ending with the
$C_{14}^{b}$-labelling in the last cycle, then applying Lemma 1.2 we obtain a
labelling with $0$ in the $d=7$ position.

Case 2: Consider $kC_{14}$ to be an arbitrary snake with $0$ in the $d=6$
position in the last cycle with $k\geq 3$ and $k$ odd. The proof is similar to
the proof of Case 1. Add $7$ to each vertex label instead of subtracting each
vertex label from $14k-7$ and apply the labelling $C_{14}^{a}$ to the final
cycle instead of $C_{14}^{b}$, with the cut vertex receiving label $7$ instead
of $6$. Then this labelling of $kC_{14}$ has all vertices labelled from
$\left[0,14k+1\right]\setminus\left\\{3,14k\right\\}$, and the edge labels are
exactly $\left[1,14k+1\right]\setminus\left\\{14k\right\\}$, with no repeated
vertex or edge label. Thus, we obtain a near graceful labelling of $kC_{14}$
and $14k+1$ in the $d=7$ position. $\Box$

Recall that a $kC_{t}$ is linear if all entries in its string are equal to
$\left\lfloor\frac{t}{2}\right\rfloor$. So, based on the results of Theorems
3.2 and 3.3 we now state a corollary for linear $kC_{10}$ and $kC_{14}$,
following the style of Theorem 1.6.

######  Corollary 3.4.

If $k\geq 1$ then the linear $kC_{10}$ and linear $kC_{14}$ are graceful if
$k$ is even and nearly graceful if $k$ is odd.

##  4 Discussion

In this paper we (near) gracefully labelled several type of snakes. In section
2, we presented a new sufficient condition which when satisfied shows there is
a graceful labelling of a $kC_{4n}$ for any string. By using a complete
$s,t$-useful cycle set we proved that if there is a complete $s,4n$-useful
cycle set with $s\geq 1$, then there exists a graceful labelling of any
$kC_{4n}$. We used the results in [5] with our results for $kC_{4n}$ and
proved that a graceful labelling exists for particular $kC_{4n}$ with string
$\left(d_{1},d_{2},\ldots,d_{k-2}\right)$, where
$d_{i}\in\left\\{2,3,4\right\\}$. Expanding these results for any $n$ and $d$
is possible but hard to apply for large $n$. We extended our main result to
the case of cyclic snakes with cycles of varying sizes. Further, we extended
the results in Theorems 1.6 to 1.9 on (near) gracefully labelled $kC_{n}$
where $n=6,8,12,16,20,24$ for all possible snakes.

As we discussed in Section 3, new approaches must be found to gracefully label
$kC_{4n+2}$ snakes, even for fixed $n$. Our collections of ad hoc methods work
to give classes for fixed $n$, but do not seem to generalize, even for “nice”
subfamilies, such as linear snakes. Thus we pose the following open question.

Question: Can we (near) gracefully label every $kC_{m}$ with $k\geq 1$ and
$m\equiv 2\ ($mod$\ 4)$?

In fact the technique we used is more general than indicated in our theorems.
Suppose we have a gracefully labelled bipartite graph $G=K_{3,4}$ as in Figure
3. If we add $3$ to each vertex label and use the cycle $H=C_{8}^{2}$ from
Table 1 then we obtain a new gracefully labelled graph as in Figure 3. Thus we
can in several cases gracefully label new graphs.

######  Theorem 4.1.

If $G$ is graceful and $H$ is a $kC_{4n}$ with $1\leq n\leq 6$, then the graph
$GH^{\ast}$ obtained by identifying any vertex in $G$ that can be labelled $0$
in some graceful labelling with any vertex in the first cycle of $H$ is
graceful.

$12$$8$$4$$3$$2$$1$$0$ Figure 2: Gracefully labelled $K_{3,4}$.

$15$$11$$7$$6$$5$$4$$3$$20$$0$$19$$1$$17$$2$$16$ Figure 3: Gracefully labelled
$GH^{\ast}$.

## References

  * [1] C. Barrientos, Graceful labellings of cyclic snakes, Ars. Combin., 60 (2001), 85–96.
  * [2] J. A. Gallian, A Dynamic Survey of Graph Labelling, Electron. J. Combin., 5 (1998), 6–34.
  * [3] R. B. Gnanajothi, Topics in Graph Theory, Ph.D. Thesis, Madurai Kamaraj University, 1991.
  * [4] D. Moulton, Graceful Labellings of Triangular Snakes, Ars. Combin., 28 (1989), 2–13.
  * [5] A. Rosa, On certain valuations of the vertices of a graph, Theory of Graphs (Internat. Symposium, Rome, July 1966), Gordon and Breach, N. Y. and Dunod Paris (1967), 349–355.