# Perfect precise colorings of plane semiregular tilings

Manuel Joseph C. Loquias  Institute of Mathematics, University of the
Philippines Diliman, 1101 Quezon City, Philippines<EMAIL_ADDRESS>and  Rovin B. Santos  Institute of Mathematics, University of the Philippines
Diliman, 1101 Quezon City, Philippines<EMAIL_ADDRESS>
(Date: July 31, 2023)

###### Abstract.

A coloring of a planar semiregular tiling ${\mathcal{T}}$ is an assignment of
a unique color to each tile of $\mathcal{T}$. If $G$ is the symmetry group of
${\mathcal{T}}$, we say that the coloring is perfect if every element of $G$
induces a permutation on the finite set of colors. If ${\mathcal{T}}$ is
$k$-valent, then a coloring of ${\mathcal{T}}$ with $k$ colors is said to be
precise if no two tiles of ${\mathcal{T}}$ sharing the same vertex have the
same color. In this work, we obtain perfect precise colorings of some families
of $k$-valent semiregular tilings in the plane, where $k\leq 6$.

###### Key words and phrases:

semiregular tilings, hyperbolic tilings, perfect colorings, precise colorings,
triangle groups

###### 2020 Mathematics Subject Classification:

05B45,52C20

Aside from their aesthetic and mathematical appeal, symmetrically colored
tilings have been studied because of their applications in crystallography and
materials science (see [16] and references therein). Of particular interest
are perfect colorings of tilings, that is, colorings where every symmetry of
the uncolored tiling sends all tiles of a given color to tiles of the same
color [7, 9]. In addition, colorings of patterns in the hyperbolic plane have
garnered attention because of their connection with quasicrystals and
structural chemistry [4, 1, 10, 17].

The term “precise coloring” was coined in [13] to refer to a coloring of the
regular triangular tiling $(3^{n})$ using $n$ colors in which no two tiles of
the same color share a common vertex.

This paper is a continuation of work on identifying perfect precise colorings
of planar tilings ${\mathcal{T}}$ in [12, 2, 18, 15]. In Section 1, we state
general results on perfect precise colorings of ${\mathcal{T}}$. Theorem 1.6
gives a characterization of perfect or chirally perfect colorings as
partitions of ${\mathcal{T}}$ that are unions of images of certain tile
orbits. We demonstrate in Section 2 how to obtain perfect precise colorings
(with $k$ colors) of some families of plane semiregular $k$-valent tilings
where $k\leq 6$. These colorings were generated using a combinatorial approach
and are verified to be perfect or chirally perfect using Theorem 1.6. Most of
the Archimedean tilings considered in [2] fall under the more general class of
semiregular tilings examined in this work, and his findings become special
cases of results in Section 2.1, and Propositions 2.1, 2.2, and 2.4.

In Section 3, we extend the definition of precise colorings of $k$-valent
semiregular tilings to include those with $n$ colors where $n>k$. This
generalization was first considered in [19], where chirally perfect colorings
of the regular triangular tiling $(3^{n})$ using $n+1$ colors in which $n$
different colors appear at each vertex were studied. Perfect precise colorings
of a semiregular tiling where the number of colors exceed the valency may be
generated using Theorem 1.6. One may use GAP [8] to find suitable subgroups of
the symmetry group of the uncolored tiling that is needed to obtain such
colorings. This was what was done to obtain the perfect precise colorings of
the $4$-valent semiregular tilings with five colors in Figs. 16 and 17 and
with six colors in Fig. 17.

Let us recall that a _planar tiling_ refers to a set of tiles
${\mathcal{T}}=\left\\{t_{i}\right\\}_{i\in\mathbbm{N}}$ that completely cover
the plane without any gaps or overlaps. If $H$ is a subgroup of the symmetry
group $G$ of a given tiling ${\mathcal{T}}$ and $t$ is a tile of
${\mathcal{T}}$, then we define the _$H$ -orbit of $t$_ as $Ht=\left\\{ht\mid
h\in H\right\\}$. Here, we refer to $Ht$ as an $H$-orbit of ${\mathcal{T}}$.
An edge-to-edge tiling of the plane by regular polygons is called
_semiregular_ (or _uniform_) if the cyclic arrangement of polygons around all
vertices are the same. Every semiregular tiling is denoted based on the
sequential arrangement of regular polygons about a vertex. That is,
$(p_{1}.p_{2}.\cdots.p_{k})$ is the semiregular tiling wherein the polygons
occuring at any vertex are a $p_{1}$-gon, a $p_{2}$-gon$,\ldots,$ and a
$p_{k}$-gon in this order or in reverse. Since the valence of each vertex of
the semiregular tiling $(p_{1}.p_{2}.\cdots.p_{k})$ is $k$, we say that the
tiling is _$k$ -valent_. Examples of semiregular tilings in the spherical,
Euclidean, and hyperbolic planes are shown in Fig. 1. It is well known that
there are only a finite number of semiregular tilings in the spherical and
Euclidean planes, but an infinite number in the hyperbolic plane. Constructing
semiregular tilings in the hyperbolic plane was explored in [11], and more
recently in [3]. In this work, we deal with particular families of $k$-valent
semiregular tilings where $k\leq 6$.

$\begin{array}[]{ccc}\includegraphics[width=101.17755pt]{5.6.6}&\includegraphics[width=101.17755pt]{3.6.3.6}&\includegraphics[width=101.17755pt]{3.3.4.3.6}\\\
(5.6.6)&(3.6.3.6)&(3.3.4.3.6)\end{array}$

Figure 1. Some semiregular tilings of the plane.

An _$n$ -coloring_ of a semiregular planar tiling $\mathcal{T}$ is a
surjective map from (the tiles of) ${\mathcal{T}}$ to a finite set of $n$
colors. If $\mathcal{T}$ is $k$-valent, then a $k$-coloring of ${\mathcal{T}}$
is called _precise_ if all $k$ colors appear at every vertex of
${\mathcal{T}}$. A coloring of $\mathcal{T}$ is said to be _perfect_ if every
element of the symmetry group $G$ of $\mathcal{T}$ effects a permutation on
the set of colors. If $G$ contains indirect symmetries but only the direct
symmetries (orientation-preserving symmetries) of $G$ permute the colors in a
coloring of $\mathcal{T}$, then we say that the coloring is _chirally
perfect_. A coloring of $\mathcal{T}$ that has only one orbit of colors under
the action of $G$ is called _transitive_. The symmetry group of every tiling
considered in this study is a triangle group ${\ast}pqr=\langle
P,Q,R:P^{2}=Q^{2}=R^{2}=(QR)^{p}=(RP)^{q}=(PQ)^{r}=1\rangle$, or a subgroup of
this group.

##  1\. Methods to Obtain Perfect Precise Colorings

There are two ways to obtain a perfect precise coloring of a tiling. The first
approach is to ensure at the outset that the coloring obtained is precise,
while making sure that the colors are permuted by generators of the symmetry
group of the tiling. This is the approach used in [2] to obtain all perfect
precise colorings of Euclidean semiregular tilings where the number of colors
is equal to the valency of the tiling. This method entails a combinatorial
case-to-case analysis, and becomes quite involved as the valency increases.
Nevertheless, this method is convenient especially for $3$-valent and
$4$-valent tilings.

For the $3$-valent case, the following lemma appears in [2].

######  Lemma 1.1.

Let $\mathcal{T}$ be a 3-valent semiregular planar tiling. If a tile of
$\mathcal{T}$ has an odd number of edges, then there is no precise coloring of
$\mathcal{T}$.

The proof of Lemma 1.1 is immediate: if the tile $t$ with an odd number of
edges has color 1, then the tiles surrounding $t$ need to be colored
alternately by colors 2 and 3, which cannot happen because of the odd number
of edges of $t$.

Let us consider a semiregular planar tiling ${\mathcal{T}}$ of the form
$(p_{1}.p_{2}.\cdots.p_{k})$ where $k\geq 3$ and the $p_{i}$’s are distinct.
Hence, ${\mathcal{T}}$ is a $k$-valent tiling with $k$ different polygons
meeting at every vertex of ${\mathcal{T}}$. Clearly, assigning a unique color
to each of the $k$ types of polygons is the only way to obtain a precise
coloring of ${\mathcal{T}}$. Since symmetries of ${\mathcal{T}}$ are
isometries, they can only map a polygonal tile to another tile of the same
type. This implies that every symmetry of ${\mathcal{T}}$ fixes each color in
the coloring of ${\mathcal{T}}$, and so the colored tiling is perfect. We
write this into a lemma for easy reference in the next section. As far as we
can tell, Lemma 1.2 does not appear in the literature.

######  Lemma 1.2.

The semiregular planar tiling $(p_{1}.p_{2}.\cdots.p_{k})$ where $k\geq 3$ and
the $p_{i}$’s are distinct has exactly one perfect precise coloring.

Observe that Lemmas 1.1 and 1.2 imply that there is no semiregular planar
tiling $(p.q.r)$ such that $p$, $q$, and $r$ are distinct and one among them
is odd. This result can be verified in [11].

Another approach is to first obtain perfect colorings of a tiling
${\mathcal{T}}$, and then discard those that are not precise. In this
approach, one may use suitable subgroups of the symmetry group of
${\mathcal{T}}$ to obtain perfect colorings of ${\mathcal{T}}$. A group-
theoretic approach to obtain perfect colorings of tilings can be traced back
to [5, 14]. The method described here is based on the work started in [4].

If there is only one orbit of tiles under the symmetry group of
${\mathcal{T}}$, then the following theorem (with $X=\mathcal{T}$ and $H=G$)
characterizes perfect colorings of ${\mathcal{T}}$ as partitions of
${\mathcal{T}}$ obtained from the orbit of a tile in ${\mathcal{T}}$ by some
subgroup of the symmetry group of ${\mathcal{T}}$ (cf. [4, 1]). The theorem is
applicable not only to perfect colorings but also to chirally perfect
colorings. The statement and proof of the theorem appears in [6], but for
completeness we give here the proof.

######  Theorem 1.3.

Let $X$ be a subset of a planar tiling ${\mathcal{T}}$ such that the subgroup
$H$ of the symmetry group $G$ of ${\mathcal{T}}$ acts transitively on $X$.
Then the $m$ colors of a coloring of $X$ are permuted by elements of $H$ if
and only if the coloring corresponds to a partition of $X$ of the form

(1) $\\{h(Jt)\mid h\in H\\},$

where $t\in X$ and $J$ is a subgroup of index $m$ in $H$ that contains
$\operatorname{Stab}_{H}(t)$ (the stabilizer of $t$ with respect to $H$).

###### Proof.

Consider a coloring of $X$ with colors $1,\ldots,m$ such that the colors are
permuted by elements of $H$. Then the group $H$ acts on the set of colors.
Take any tile $t\in X$ and assume that its color is $1$. Let $J$ be the
stabilizer of $1$ with respect to $H$. Since $H$ acts transitively on $X$, it
also acts transitively on the set of $m$ colors. It then follows from the
Orbit-Stabilizer theorem that $[H:J]=m$. Note that every element of $H$ that
fixes $t$ must also fix color $1$, and so $\operatorname{Stab}_{H}(t)\subseteq
J$. Finally, the coloring of $X$ corresponds to the partition $\\{h(Jt)\mid
h\in H\\}$ of $X$, where the color of the tiles in $h(Jt)$ is the color of
tile $ht$.

Conversely, $\cup_{h\in H}(h(Jt))=X$ since $H$ acts transitively on $X$.
Moreover, because the stabilizer of $t$ in $H$ is in $J$, the $m$ sets in
$\mathcal{C}=\\{h(Jt)\mid h\in H\\}$ are pairwise disjoint. It is clear that
$\mathcal{C}$ is invariant under $H$, which implies that the elements of $H$
permute the colors in a coloring of $X$ corresponding to $\mathcal{C}$. ∎

Observe from the proof of Theorem 1.3 that the subgroup $J$ in (1) consists of
elements of $H$ that fix the color of tile $t$, or equivalently, $J$ is the
stabilizer of the set $Jt$ under the action of $H$ on the elements of the
partition of $X$ in (1).

We now look at results on how to perfectly color a planar tiling that has more
than one transitivity class under the action of its symmetry group. While the
basic ideas are similar to those discussed in [9] concerning perfect
colorings, the succeeding results extend beyond perfect colorings to include
chirally perfect colorings as well. The next lemma bears resemblance to Lemma
3 of [9].

######  Lemma 1.4.

Let ${\mathcal{T}}$ be a planar tiling with symmetry group $G$ and $H$ be a
subgroup of $G$. If the $H$-orbits $X_{i}$ and $X_{j}$ of ${\mathcal{T}}$
share some color in a coloring of ${\mathcal{T}}$ where each element of $H$
permutes the colors, then $X_{i}$ and $X_{j}$ have the same set of colors. In
particular, $X_{i}$ and $X_{j}$ have the same number of colors.

###### Proof.

Suppose that the tiles in $X_{i}$ have colors $1,2,\ldots,m$, and let $X_{i}$
and $X_{j}$ share color $1$. Without loss of generality, assume that tiles
$t_{i}\in X_{i}$ and $t_{j}\in X_{j}$ have color 1. Since the group $H$ acts
transitively on $X_{i}$, there exist $h_{1},\ldots,h_{m}\in H$ such that the
color of tile $h_{k}t_{i}$ is $k$ for $1\leq k\leq m$. Since the colors are
permuted by elements of $H$, the color of tile $h_{k}(t_{j})$ should also be
$k$ for each $1\leq k\leq m$. Hence all colors that appear in $X_{i}$ occur in
$X_{j}$. The same argument reversed yields the claim. ∎

Let $G$ be the symmetry group of a planar tiling ${\mathcal{T}}$. The next
lemma (with $O={\mathcal{T}}$ and $H=G$) deals with the instance when the same
set of colors is used to color each orbit of ${\mathcal{T}}$ under $G$ to come
up with a perfect coloring of ${\mathcal{T}}$. This result aligns closely to
Theorem 4 of [9].

######  Lemma 1.5.

Let $O$ be a subset of a planar tiling ${\mathcal{T}}$ with symmetry group
$G$, and $H$ a subgroup of $G$. If all the H-orbits $X_{1},\ldots,X_{n}$ of
$O$ have the same set of $m$ colors in a coloring of $O$, then the colors of
$O$ are permuted by elements of $H$ if and only if the coloring corresponds to
a partition of $O$ of the form $\\{h(Jt_{1}\cup\cdots\cup Jt_{n})\mid h\in
H\\}$, where $t_{i}\in X_{i}$, $1\leq i\leq n$, $J$ is a subgroup of index $m$
in $H$ that contains
$\operatorname{Stab}_{H}(t_{1}),\ldots,\operatorname{Stab}_{H}(t_{n})$.

###### Proof.

It follows from Theorem 1.3 that the coloring of $X_{1}$ can be viewed as the
partition

$\\{h(Jt_{1})\mid h\in H\\}$

for some tile $t_{1}\in X_{1}$ with color $c(t_{1})$ and
$J=\operatorname{Stab}_{H}(c(t_{1}))$ of index $m$ in $H$ that contains
$\operatorname{Stab}_{H}(t_{1})$. For $i\in\\{2,\ldots,n\\}$, choose $t_{i}\in
X_{i}$ such that tile $t_{i}$ has color $c(t_{1})$. Since elements of $H$
permute the colors in the coloring of $O$, each coloring of $X_{i}$
corresponds to the partition $\\{h(Jt_{i})\mid h\in H\\}$ and
$\operatorname{Stab}_{H}(t_{i})$ is contained in $J$ for $2\leq i\leq n$. This
shows that the coloring of $O$ corresponds to the partition
$\\{h(Jt_{1}\cup\cdots\cup Jt_{n})\mid h\in H\\}$ of $O$, where the color of
the tiles in $h(Jt_{1}\cup\cdots\cup Jt_{n})$ is the color of tile $ht_{1}$.

For the other direction, it follows from Theorem 1.3 that the $m$ colors in
the coloring of each orbit $X_{i}$ are permuted by elements of $H$. In
addition, $J$ is the stabilizer of the color of tiles $t_{1},\ldots,t_{n}$,
that is, $J$ is the stabilizer of the subsets $Jt_{1},\ldots,Jt_{n}$ under the
action of $H$ on the respective partitions of $X_{1},\ldots,X_{n}$. This
implies that the action of $H$ on the set of colors in the coloring of each
orbit $X_{i}$ are equivalent. Hence, the elements of $H$ permute the colors in
the resulting coloring of $O$. ∎

Finally, given a colored tiling ${\mathcal{T}}$ whose colors are permuted by
the subgroup $H$ of the symmetry group $G$ of ${\mathcal{T}}$, we may group
the orbits of ${\mathcal{T}}$ under $H$ having the same set of colors by Lemma
1.4. Upon doing this, we apply Lemma 1.5 to associate the coloring of each
collection of orbits as a partition of the union of the orbits. This yields
the following general result, which is more straightforward and easier to
apply compared to Theorem 5 of [9].

######  Theorem 1.6.

Let $H$ be a subgroup of the symmetry group $G$ of a planar tiling
${\mathcal{T}}$, and suppose $X_{1},\ldots,X_{n}$ are the $H$-orbits of
${\mathcal{T}}$. Then the $m$ colors in a coloring of ${\mathcal{T}}$ are
permuted by elements of $H$ if and only if it corresponds to a partition of
${\mathcal{T}}$ of the form $\cup_{i\in I}{O_{i}}$, where

(2) $O_{i}=\bigg{\\{}h\Big{(}\bigcup_{j=1}^{n_{i}}J_{i}t_{i,j}\Big{)}\mid h\in
H\bigg{\\}},$

with $\sum_{i\in I}{n_{i}}=n$, $\cup_{i\in I}\cup_{j=1}^{n_{i}}{t_{i,j}}$ is a
complete set of $H$-orbit representatives of ${\mathcal{T}}$, $J_{i}$ is a
subgroup of $H$ that contains $\operatorname{Stab}_{H}(t_{i,j})$, for $1\leq
j\leq n_{i}$, and $m=\sum_{i\in I}[H:J_{i}]$.

Hence, we may use Theorem 1.6 to obtain all (chirally) perfect colorings of an
$m$-valent semiregular planar tiling ${\mathcal{T}}$. To reduce the number of
non-precise perfect colorings that will be obtained using this method, vertex
and edge adjacencies of tiles may be considered a priori. For instance, if $k$
tiles in the subset $O_{i}$ of ${\mathcal{T}}$ meet at a vertex, then the
coloring of $O_{i}$ must have at least $k$ colors, or equivalently, the
subgroup $J_{i}$ of $H$ must satisfy $[H:J_{i}]\geq k$.

It can be shown that this procedure yields the same results in [2] for
(chirally) perfect colorings of semiregular Euclidean planar tilings. Note
though that in this approach, knowledge of the subgroup structure of the
symmetry groups of the tilings plays an important role. This makes the method
applicable for specific tilings, but difficult to use when obtaining results
for a family of tilings.

To illustrate, consider the $(3.4.6.4)$ tiling ${\mathcal{T}}$ of the
Euclidean plane with symmetry group $G$ of type $\ast 632$. The tiling
${\mathcal{T}}$ has three orbits under the action of $G$: the orbit $X_{1}$ of
hexagons, the orbit $X_{2}$ of squares, and the orbit $X_{3}$ of triangles. It
has only one perfect precise coloring with four colors. This coloring consists
of two orbits of colors: the orbit $O_{1}=X_{1}\cup X_{2}$, where $X_{1}$ and
$X_{2}$ share the same set of three colors, while each tile in the orbit
$O_{2}=X_{3}$ is assigned the same color (see Fig. 3 of [2]). The coloring of
$O_{1}$ corresponds to the partition $\\{g(Jt_{1}\cup Jt_{2})\mid g\in G\\}$,
where $t_{1}$ is a hexagon, $t_{2}$ is a square that is not adjacent to any
tile in $Jt_{1}$, and $J$ is a subgroup of index 3 in $G$ of type $\ast 632$
that contains $\operatorname{Stab}_{G}(t_{1})=D_{6}$ and
$\operatorname{Stab}_{G}(t_{2})=D_{2}$. On the other hand, the trivial
coloring of $O_{2}$ corresponds to the partition $\\{g(Gt_{3})\mid g\in
G\\}=\\{X_{3}\\}$, where $t_{3}$ is any triangle in ${\mathcal{T}}$.

##  2\. Examples of Perfect Precise Colorings

In this section, we obtain perfect precise colorings of some semiregular
planar tilings ${\mathcal{T}}$ of valencies 3, 4, 5, and 6. We restrict the
number of colors in the colorings of ${\mathcal{T}}$ to be the valency of
${\mathcal{T}}$. To obtain these colorings, we start out with a combinatorial
argument to exhaust all possible perfect precise colorings. Afterwards, we
verify that the obtained colorings of ${\mathcal{T}}$ are indeed perfect by
giving the corresponding partition of ${\mathcal{T}}$ of the form given in
(2). We note that if the coloring of ${\mathcal{T}}$ is perfect and there is a
vertex $v$ of ${\mathcal{T}}$ such that no two tiles meeting at $v$ have the
same color, then the coloring of ${\mathcal{T}}$ must be precise. This is
because the action of the symmetry group of ${\mathcal{T}}$ is transitive on
the set of vertices of ${\mathcal{T}}$.

###  2.1. 3-valent Semiregular Planar Tilings

We have the following result: a 3-valent semiregular tiling
${\mathcal{T}}=(p_{1}.p_{2}.p_{3})$ admits exactly one perfect precise
coloring if and only if each $p_{i}$ is even. Indeed, the forward direction
follows from Lemma 1.1. For the converse, observe that assigning a tile $t$ by
color 1, and then assigning colors 2 and 3 alternately to the surrounding
tiles of $t$ gives rise to a precise coloring of ${\mathcal{T}}$. It remains
to show that this unique precise coloring of ${\mathcal{T}}$ is perfect. One
way to conclude this is to look at the effects of the mirror reflections $P$,
$Q$, and $R$ on each of the three colors. More formally, we note that the case
where $p_{1}$, $p_{2}$, and $p_{3}$ are distinct is covered by Lemma 1.2. The
case when ${\mathcal{T}}$ is regular, that is, when $p_{1}=p_{2}=p_{3}=p$ for
some $p$, was considered in [15]. In this case, the symmetry group of
${\mathcal{T}}$ is $G=\ast p32$ and the precise coloring of ${\mathcal{T}}$ is
perfect because it corresponds to the partition $\left\\{g(Jt)\mid g\in
G\right\\}$ of ${\mathcal{T}}$, where $t$ is the $p$-gon stabilized by the
$p$-fold rotation $QR$ and $J$ is the subgroup $\langle PRQRP,Q,R\rangle$ of
index $3$ in $G$. The last case is, without loss of generality, when
$q:=p_{1}$ is distinct from $2p:=p_{2}=p_{3}$ for some integer $p$. Here
$G=\ast pq2$ and the precise coloring of ${\mathcal{T}}$ is perfect because it
corresponds to the partition $\left\\{Gt_{1}\right\\}\cup\left\\{g(Jt_{2})\mid
g\in G\right\\}$ of ${\mathcal{T}}$, where $t_{1}$ is any $q$-gon, $t_{2}$ is
the $2p$-gon stabilized by the $p$-fold rotation $QR$, and $J$ is the subgroup
$\langle PRP,Q,R\rangle$ of index 2 in $G$. Fig. 2 illustrates the perfect
precise 3-colorings of some 3-valent semiregular tilings, where the axes of
the generators $PRP$, $Q$, and $R$ of the subgroup $J$ of $G$ are shown on the
coloring of the $(8.6.6)$ tiling.

$\begin{array}[]{ccc}\includegraphics[height=101.17755pt]{10.4.4}&\includegraphics[height=101.17755pt]{4.6.12}&\includegraphics[height=101.17755pt]{866ppc}\\\
(10.4.4)&(4.6.12)&(8.6.6)\\\ \end{array}$

Figure 2. Perfect precise 3-coloring of some 3-valent semiregular tilings.

###  2.2. 4-valent Semiregular Planar Tilings

We consider first the family of semiregular planar tilings
${\mathcal{T}}_{1}=(p.q.p.q)$. The tiling ${\mathcal{T}}_{1}$ consists of two
orbits of tiles, the set of $p$-gons and the set of $q$-gons, under its
symmetry group $G={\ast}pq2$. We note the following: (1) the center of each
$p$-gon ($q$-gon) is a center of a $p$-fold ($q$-fold) rotation; (2) $p$
$q$-gons are adjacent to every $p$-gon ($q$ $p$-gons are adjacent to every
$q$-gon); and (3) $p$ $p$-gons are attached to each $p$-gon at the vertices
($q$ $q$-gons are attached to each $q$-gon at the vertices). Using these
properties of ${\mathcal{T}}_{1}$, we now outline how a perfect precise
4-coloring of ${\mathcal{T}}_{1}$ is achieved.

Let $t$ be any $p$-gon in the tiling ${\mathcal{T}}_{1}$ and say $t$ is given
color 1. We then assign colors 2, 3, and 4 to the $p$-gons and $q$-gons that
surround $t$. There are two ways to assign them to the $p$ $q$-gons that are
adjacent to $t$: (1) the $q$-gons alternately get two of the three colors, say
colors $2$ and $3$ (see Fig. 3(a)), or (2) the $q$-gons get colors $2$, $3$,
and $4$ in a cyclical manner (see Fig. 3(b)). We deal with these two cases
separately.

$\begin{array}[]{cc}\includegraphics[width=86.72377pt]{pqpqcase1new}&\includegraphics[width=86.72377pt]{pqpqcase2new}\\\
(a)&(b)\\\ \end{array}$

Figure 3. Possible color assignments for a perfect precise 4-coloring of the
tiling $(p.q.p.q)$.

If the $p$ $q$-gons adjacent to $t$ are assigned colors $2$ and $3$
alternately, then $p$ must be even for the resulting coloring of
${\mathcal{T}}_{1}$ to be precise. This assignment forces all $p$ $p$-gons
attached to $t$ to be assigned color $4$. Let us now look at the colored
$q$-gons adjacent to $t$. Observe that each of these $q$-gons has a $q$-fold
rotational symmetry that sends $t$ (colored 1) to one of the $p$-gons colored
$4$ and sends a $p$-gon colored $4$ to $t$. Since we aim for the coloring of
${\mathcal{T}}_{1}$ to be perfect, the $q$ $p$-gons attached to each colored
$q$-gon must have alternating colors $1$ and $4$. To ensure that the coloring
of ${\mathcal{T}}_{1}$ remains precise, $q$ must also be even. In addition the
$q$ $q$-gons attached to a $q$-gon colored $2$ should be assigned color $3$,
and all $q$ $q$-gons attached to a $q$-gon colored $3$ should be assigned
color $2$. Applying the same arguments successively to each layer of $p$-gons
and $q$-gons surrounding $t$, we obtain a perfect precise $4$-coloring of
${\mathcal{T}}_{1}$ where the $q$-gons adjacent to each $p$-gon are
alternately colored $2$ and $3$ and the $p$-gons adjacent to each $q$-gon are
alternately colored $1$ and $4$. The coloring consists of two color orbits and
is precise by construction. The coloring is indeed perfect because it
corresponds to the partition $O_{1}\cup O_{2}$ of ${\mathcal{T}}_{1}$, with
$O_{1}=\left\\{g(J_{1}t_{1})\mid g\in G\right\\}$ where $t_{1}$ is the $p$-gon
stabilized by the $p$-fold rotation $QR$ and $J_{1}=\langle PRP,Q,R\rangle$ of
index $2$ in $G$ and $O_{2}=\left\\{g(J_{2}t_{2})\mid g\in G\right\\}$ where
$t_{2}$ is the $q$-gon stabilized by the $q$-fold rotation $PR$ and
$J_{2}=\langle P,QRQ,R\rangle$ of index $2$ in $G$.

On the other hand, if the $p$ $q$-gons that are adjacent to $t$ are colored
$2$, $3$, and $4$ in a repeating manner, then $p$ must be a multiple of three
for the resulting coloring of ${\mathcal{T}}_{1}$ to be precise. This color
assignment implies that the $p$ $p$-gons attached to $t$ should also be
colored $2$, $3$, and $4$ in a repeating manner such that the $p$-gon adjacent
to $q$-gons colored $2$ and $3$ (resp. $2$ and $4$, and $3$ and $4$) should be
colored $4$ (resp. $3$ and $2$). We now turn our attention to the colored
$q$-gons attached to $t$. If the $q$-gon $t^{\prime}$ is colored $2$, then it
will be surrounded by three adjacent $p$-gons with colors $4$, $1$, and $3$.
For the coloring of ${\mathcal{T}}_{1}$ to be perfect, the $q$ $p$-gons
adjacent to $t^{\prime}$ need to be colored $1$, $3$, and $4$ in a repeating
manner. This means that $q$ must also be a multiple of $3$. At the same time,
the $q$ $q$-gons attached to $t^{\prime}$ should be colored $1$, $3$, and $4$
in a repeating manner so that the coloring of ${\mathcal{T}}_{1}$ remains
precise. Similar arguments yield the colors of every $p$-gon and $q$-gon
surrounding each colored $q$-gon adjacent to $t$. Repeating the same reasoning
to each layer of $p$-gons and $q$-gons surrounding $t$ results to a perfect
precise $4$-coloring of ${\mathcal{T}}_{1}$ where the $p$-gons and $q$-gons
surrounding each $p$-gon and each $q$-gon are colored by the other three
colors in a cyclical manner. The symmetry group $G$ of ${\mathcal{T}}_{1}$
acts transitively on the four colors, and the coloring of ${\mathcal{T}}_{1}$
is precise by construction. The coloring of ${\mathcal{T}}_{1}$ is also
perfect because it corresponds to the partition $\left\\{g(Jt_{1}\cup
Jt_{2})\mid g\in G\right\\}$ where $t_{1}$ is the $p$-gon stabilized by the
$p$-fold rotation $QR$, $t_{2}$ is the $q$-gon stabilized by the reflection
$PRQRQRP$, and $J=\langle PRPRPR,PRQRQRP,Q,R\rangle$ of index $4$ in $G$.

Thus, we obtain the following result.

######  Proposition 2.1.

Let ${\mathcal{T}}$ be the semiregular planar tiling $(p.q.p.q)$. Then the
number of perfect precise $4$-colorings of ${\mathcal{T}}$ is

  1. (1)

one if and only if one of the following hold: $p$ and $q$ are even and $p$ or
$q$ is not a multiple of three, or $p$ and $q$ are multiples of 3 and $p$ or
$q$ is not even; and

  2. (2)

two if and only if $p$ and $q$ are divisible by 6.

Proposition 2.1 remains valid even if $p=q$ (in [15], the unique perfect
precise $4$-colorings of the octahedron $(3^{4})$ and the regular Euclidean
tiling $(4^{4})$ are discussed, and the two perfect precise $4$-colorings of
the regular hyperbolic tiling $(6^{4})$ are shown). The existence of the
unique perfect precise 4-coloring of the Euclidean semiregular tiling
$(6.3.6.3)$ in [2] also agrees with Proposition 2.1. Note that Proposition 2.1
implies that there are no perfect precise 4-colorings of the semiregular
spherical tilings $(3.4.3.4)$ and $(3.5.3.5)$. We show in Fig. 4 perfect
precise 4-colorings of the semiregular tiling $(p.q.p.q)$ for some values of
$p$ and $q$. The axes of the generators of the subgroups $J_{1}=\langle
PRP,Q,R\rangle$ and $J_{2}=\langle P,QRQ,R\rangle$ of $G$ are drawn on the
coloring of the $(6.4.6.4)$ tiling.

$\begin{array}[]{ccc}\includegraphics[height=101.17755pt]{4646ppc}&\includegraphics[height=101.17755pt]{4848ppc}&\includegraphics[height=101.17755pt]{6969ppc}\\\
(6.4.6.4)&(8.4.8.4)&(9.6.9.6)\\\
\includegraphics[height=101.17755pt]{3939ppc}&\includegraphics[height=101.17755pt]{126126ppc1}&\includegraphics[height=101.17755pt]{126126ppc2}\\\
(9.3.9.3)&(6.12.6.12)_{1}&(6.12.6.12)_{2}\end{array}$

Figure 4. Some perfect precise 4-colorings of the tiling $(p.q.p.q)$.

Another semiregular tiling with symmetry group $G={\ast}pq2$ is the tiling
${\mathcal{T}}_{2}=(p.4.q.4)$, where $p\neq q$. There is a $p$-fold ($q$-fold)
rotation symmetry of ${\mathcal{T}}_{2}$ about the center of each $p$-gon
($q$-gon), and a $2$-fold rotation symmetry about the center of every square.
The action of $G$ on ${\mathcal{T}}_{2}$ forms at most 3 orbits of tiles: the
set of squares, the set of $p$-gons, and the set of $q$-gons. No two $p$-gons
($q$-gons) in ${\mathcal{T}}_{2}$ share a common vertex nor a common edge, and
each $p$-gon ($q$-gon) is surrounded by $p$ ($q$) adjacent squares.

We use a similar approach as for the tiling ${\mathcal{T}}_{1}$ to obtain
perfect precise 4-colorings of the tiling ${\mathcal{T}}_{2}$. We start with
any $p$-gon $t$ in ${\mathcal{T}}_{2}$ and assign it the color $1$. There are
two ways to assign colors to the $p$ squares adjacent to $t$: alternating
using two colors, say $2$ and $3$, or cyclically using the three colors $2$,
$3$, and $4$. The color assignment of the squares will force the color
assignment of the $p$ $q$-gons around $t$. Afterwards, we color the $q$
$p$-gons and $q$ squares surrounding each $q$-gon around $t$. If the $q$-gon
$t^{\prime}$ attached to $t$ has color $2$ (resp. $3$ and $4$), then there are
two possibilities: assign color $1$ to every $p$-gon attached to $t^{\prime}$
and assign colors $3$ and $4$ (resp. $2$ and $4$, and $2$ and $3$) alternately
to the squares around $t^{\prime}$, or assign colors $1,3,4$ (resp. $1,2,4$
and $1,2,3$) to the $p$-gons and squares in some repeating manner. The color
assignment of the succeeding layers of $p$-gons, $q$-gons, and squares around
$t$ are then determined inductively. We list the possibilities below and
summarize them in the proposition thereafter.

  1. (i)

If $p$ is even, then the squares around $t$ are colored $2$ and $3$ in
succession while every $q$-gon attached to $t$ are colored $4$.

If $q$ is even, then the $p$-gons surrounding each $q$-gon attached to $t$ are
colored $1$ while the surrounding squares are colored $2$ and $3$ in
succession (see Fig. 5(a)). In this case, the perfect precise $4$-coloring of
${\mathcal{T}}_{2}$ corresponds to the partition $O_{1}\cup O_{2}\cup O_{3}$,
with $O_{1}$ the set of $p$-gons, $O_{2}$ the set of $q$-gons, and
$O_{3}=\left\\{g(Jt)\mid g\in G\right\\}$ where $t$ is the square stabilized
by the $2$-fold rotation $PQ$ and $J=\langle P,Q,RPR,RQR\rangle$ of index $2$
in $G$.

If $q$ is a multiple of $3$ then the $p$-gons and squares surrounding each
$q$-gon attached to $t$ are colored $1,2,3$ in some repeating manner (see Fig.
5(b)). Here, the perfect precise $4$-coloring of ${\mathcal{T}}_{2}$
corresponds to the partition $O_{1}\cup O_{2}$, with $O_{1}$ the set of
$q$-gons and $O_{2}=\left\\{g(Jt_{1}\cup Jt_{2})\mid g\in G\right\\}$ where
$t_{1}$ is the $p$-gon stabilized by the $p$-fold rotation $QR$, $t_{2}$ is
the square stabilized by the reflection $PRPRP$, and $J=\langle
PRPRP,PRQRP,Q,R\rangle$ of index $3$ in $G$.

  2. (ii)

If $p$ is a multiple of 3 and $q$ is even, then interchanging the values of
$p$ and $q$ yields the case above.

The case where $p$ and $q$ are odd multiples of $3$ does not give rise to a
perfect precise $4$-coloring of ${\mathcal{T}}_{2}$. Indeed, since $p$ is an
odd multiple of $3$, then the $p$ squares and $p$ $q$-gons around $t$ can only
be colored $2,3,4$ in a cyclical fashion for the coloring to be precise.
Consider a $q$-gon $t^{\prime}$ attached to $t$ with color $4$. We can only
color the $p$-gons and squares around $t^{\prime}$ by $1,2,3$ in some
repeating manner because $q$ is an odd multiple of $3$ (see Fig. 5(c)). This
will mean that the second $p$-gon adjacent to a square colored $2$ will be
assigned color $3$. However, the same $p$-gon is attached to a $q$-gon colored
3, and thus the coloring obtained will not be precise.

$\begin{array}[]{ccc}\includegraphics[width=86.72377pt]{p4q4case1new}&\includegraphics[width=86.72377pt]{p4q4case2new}&\includegraphics[width=86.72377pt]{p4q4case4new}\\\
(a)&(b)&(c)\end{array}$

Figure 5. Possible color assignments for a perfect precise 4-coloring of the
tiling $(p.4.q.4)$.

######  Proposition 2.2.

Let ${\mathcal{T}}$ be the semiregular tiling $(p.4.q.4)$ with $p\neq q$. Then
the number of perfect precise $4$-colorings of ${\mathcal{T}}$ is

  1. (1)

one if and only if $p$ and $q$ are even and neither is a multiple of 3, or one
of $p$ and $q$ is even and the other is an odd multiple of 3;

  2. (2)

two if and only if $p$ and $q$ are even and exactly one of them is a multiple
of 3; and

  3. (3)

three if and only if $p$ and $q$ are multiples of 6.

Proposition 2.2 is consistent with the unique perfect precise 4-coloring of
the Archimedean tiling $(3.4.6.4)$ obtained in [2]. The perfect precise
4-colorings of the semiregular hyperbolic tilings $(p.4.q.4)$ for some values
of $p$ and $q$ in Fig. 6 illustrates Proposition 2.2. The axes of the
generators of the corresponding subgroups $J$ of $G$ are shown for the two
distinct perfect precise $4$-colorings of the $(8.4.6.4)$ tiling. The
semiregular spherical tiling $(5.4.3.4)$ admits no perfect precise coloring
with 4 colors.

$\begin{array}[]{ccc}\includegraphics[height=108.405pt]{4348ppc}&\includegraphics[height=108.405pt]{4648ppc1}&\includegraphics[height=108.405pt]{4648ppc2}\\\
(8.4.3.4)&(8.4.6.4)_{1}&(8.4.6.4)_{2}\\\
\includegraphics[height=108.405pt]{46412ppc1}&\includegraphics[height=108.405pt]{46412ppc3}&\includegraphics[height=108.405pt]{46412ppc2}\\\
(12.4.6.4)_{1}&(12.4.6.4)_{2}&(12.4.6.4)_{3}\end{array}$

Figure 6. Some perfect precise 4-colorings of the tiling $(p.4.q.4)$.

###  2.3. 5-valent Semiregular Planar Tilings

We now determine the perfect and chirally perfect precise 5-colorings of the
family of 5-valent tilings ${\mathcal{T}}=(3.3.p.3.q)$. If $p\neq q$, then the
symmetry group of ${\mathcal{T}}$ is $G=pq2$. The tiling ${\mathcal{T}}$ has a
$p$-fold ($q$-fold) rotation symmetry about the center of each $p$-gon
($q$-gon), and a $2$-fold symmetry at the midpoint of the shared edge of any
two adjacent triangles. The set of $p$-gons, the set of $q$-gons, and the set
of triangles form distinct orbits under the action of $G$ on ${\mathcal{T}}$.
If $p=q$, then the symmetry group of ${\mathcal{T}}$ is $G^{\prime}=p{\ast}2$.
In this case, ${\mathcal{T}}$ has a reflection symmetry along the shared edge
of each pair of adjacent triangles. The set of $p$-gons form one orbit under
the action of $G^{\prime}$ on ${\mathcal{T}}$, but is partitioned into two
orbits under the action of the rotation group $H=pp2$. It is easy to see that
two $p$-gons that have a common vertex belong to different $H$-orbits.

First, we show two things: (a) if $p=q$, then ${\mathcal{T}}$ does not admit
any perfect precise $5$-coloring, and (b) if $p\neq q$, then every $p$-gon is
assigned the same color or every $q$-gon is assigned the same color in any
perfect precise 5-coloring of ${\mathcal{T}}$.

Let $p=q$. Assign distinct colors to five tiles around a vertex of
${\mathcal{T}}$ as shown in Fig. 7(a). Notice that ${\mathcal{T}}$ has a
reflection symmetry $\alpha$ along the red line that passes through the shared
edge of triangles $b$ and $c$. If the $5$-coloring of ${\mathcal{T}}$ is to be
perfect, then $\alpha$ must fix the color of triangle $a$. This means that
neither triangle $b$ nor $c$ should be assigned the same color as triangle
$a$, otherwise the resulting $5$-coloring of ${\mathcal{T}}$ will not be
precise. Hence, we are left with no choice but for the $p$-gon $d$ to have the
same color as triangle $a$. This yields a contradiction because $\alpha$ sends
the $p$-gon $d$ to a $p$-gon of a different color. We have now shown that it
is impossible to perfectly color ${\mathcal{T}}$ with 5 colors such that the
resulting coloring is precise whenever $p=q$.

$\begin{array}[]{cc}\includegraphics[height=72.26999pt]{33p3pnoppc}&\includegraphics[height=72.26999pt]{33p3qnotransitive}\\\
(a)&(b)\\\ \end{array}$

Figure 7. (a) Nonexistence of a perfect precise 5-coloring of the tiling
$(3.3.p.3.p)$; (b) Nonexistence of a transitive perfect precise 5-coloring of
the tiling $(3.3.p.3.q)$.

We now proceed to show that if $p\neq q$, then all $p$-gons have the same
color or all $q$-gons have the same color in a perfect precise $5$-coloring of
${\mathcal{T}}$.

Just as before, we start with five tiles meeting at a vertex $v$ that are
assigned different colors (see Fig. 7(b)). For the 5-coloring of
${\mathcal{T}}$ to be precise, the $p$-gon $c$ can be colored with either the
color of triangle $a$ or the color orange of the $p$-gon adjacent to triangle
$a$. However, the $p$-fold rotation symmetry $\beta$ about the center of the
orange $p$-gon fixes the color orange. This means that if $p$-gon $c$ is
colored orange, then every $p$-gon in the tiling should be colored orange for
the $5$-coloring of ${\mathcal{T}}$ to be perfect. On the other hand, if
$p$-gon $c$ has the same color as triangle $a$, then the $q$-gon $d$ will be
forced to have the same color brown of the $q$-gon $b$. As $\beta$ now fixes
color brown, every $q$-gon in ${\mathcal{T}}$ will be colored brown so that
the resulting $5$-coloring of ${\mathcal{T}}$ is perfect. This proves the
claim.

We now discuss how to obtain perfect precise $5$-colorings of ${\mathcal{T}}$
if $p\neq q$. We classify the triangles around a $p$-gon ($q$-gon) into two
types: those that share a common edge with the $p$-gon ($q$-gon) will be
called type I triangles while those that share only a vertex with the $p$-gon
($q$-gon) will be called type II triangles. Suppose $t$ is a $p$-gon and
$t^{\prime}$ is a $q$-gon in ${\mathcal{T}}$ having a common vertex. We assign
different colors to the tiles around this vertex, in particular, we assign
color $1$ to the $p$-gon $t$, color $2$ to the triangle adjacent to both $t$
and $t^{\prime}$, color $3$ to the $q$-gon, color $4$ to the triangle adjacent
to $t^{\prime}$ but not to $t$, and color $5$ to the remaining triangle
adjacent to $t$ but not to $t$’. From earlier, we have the following three
cases.

  1. (i)

Suppose all $p$-gons are colored $1$ and all $q$-gons are colored $3$. The
type I triangles around $t$ can be colored either $2$ and $5$ alternately or
$2,5,4$ in a cyclical manner. In the former, $p$ must be even and all type II
triangles around $t$ will be colored $4$. This color assignment yields a type
I triangle and a type II triangle around $t^{\prime}$ that are colored $4$.
Hence, the type I and type II triangles around $t^{\prime}$ must be colored
$2$, $4$, $5$ in some repeating manner which means that $q$ must be a multiple
of $3$ (see Fig. 8(a)). The colors of the layers of triangles around $t$ are
now uniquely determined, and the resulting $5$-coloring of ${\mathcal{T}}$ is
precise by construction. In addition, the $5$-coloring is perfect because it
corresponds to the partition $O_{1}\cup O_{2}\cup O_{3}$ of ${\mathcal{T}}$,
where $O_{1}$ is the set of $p$-gons, $O_{2}$ is the set of $q$-gons, and
$O_{3}=\left\\{g(Jt)\mid g\in G\right\\}$ where $t$ is any type II triangle
around $t$ and $J=\langle RQ,(PR)^{3},PRQRPQ\rangle$ of index $3$ in $G$.

If instead the type I triangles around $t$ are colored $2,5,4$ in a cyclical
manner, then similar arguments show that $p$ must be a multiple of three and
$q$ must be even for a perfect precise $5$-coloring of ${\mathcal{T}}$ to be
obtained (see Fig. 8(b)). In fact, the resulting coloring is exactly the same
coloring above where the roles of $p$ and $q$ are interchanged.

  2. (ii)

Suppose all $p$-gons are colored $1$ but not all $q$-gons are colored $3$ (see
Figs. 8(c) and (d)). The $q$-gon adjacent to the triangle colored $5$ should
be colored $2$, otherwise, the $5$-coloring of ${\mathcal{T}}$ will not be
precise or all $q$-gons will be colored $3$. This implies that the type I
triangles and $q$-gons surrounding $t$ can be cyclically colored by the three
colors $2,5,3$ or the four colors $2,5,4,3$. An analogous argument shows that
in the former, $p$ and $q$ must be multiples of $3$ and the resulting perfect
precise $5$-coloring of ${\mathcal{T}}$ corresponds to the partition
$O_{1}\cup O_{2}$ of ${\mathcal{T}}$, with $O_{1}$ the set of $p$-gons and
$O_{2}=\left\\{g(Jt_{1}\cup Jt_{2})\mid g\in G\right\\}$ where $t_{1}$ is the
$q$-gon stabilized by the $q$-fold rotation $PRQRPQRP$, $t_{2}$ is any type II
triangle around $t$, and $J=\langle RQ,PRQRPQRP,PRQRQRPQ\rangle$ of index $4$
in $G$.

If instead the type I triangles and $q$-gons around $t$ are assigned four
colors, we obtain that $p$ should be a multiple of $4$ and $q$ should be even.
The $5$-coloring of ${\mathcal{T}}$ obtained is perfect and precise that
corresponds to the partition $O_{1}\cup O_{2}$, with $O_{1}$ the set of
$p$-gons, $O_{2}=\left\\{g(Jt_{1}\cup Jt_{2})\mid g\in G\right\\}$ where
$t_{1}=t^{\prime}$, $t_{2}$ is a type I or type II triangle around $t$ with
color $3$, and $J=\langle(RQ)^{4},PR,QRPRPQ\rangle$ of index $4$ in $G$.

  3. (iii)

Suppose all $q$-gons are colored $3$ but not all $p$-gons are colored $1$ (see
Figs. 8(e) and (f)). The $p$-gon adjacent to the triangle colored $4$ should
be colored $2$, otherwise, all $p$-gons will be colored $1$. This implies that
the type I triangles and $p$-gons surrounding $t$ can be colored either $2$
and $5$ alternately or $2,5,4$ in a cyclical manner. Similar arguments show
that to obtain a perfect precise $5$-coloring of ${\mathcal{T}}$, $p$ must be
even and $q$ must be a multiple of $4$ in the former while $p$ and $q$ must be
divisible by $3$ in the latter. In both instances, the coloring generated is
the same coloring obtained from Case (ii) when the values of $p$ and $q$ are
interchanged.

$\begin{array}[]{ccc}\includegraphics[height=72.26999pt]{33p3qcase1a}&\includegraphics[height=72.26999pt]{33p3qcase1b}&\includegraphics[height=72.26999pt]{33p3qcase2a}\\\
(a)&(b)&(c)\\\
\includegraphics[height=72.26999pt]{33p3qcase2b}&\includegraphics[height=72.26999pt]{33p3qcase3a}&\includegraphics[height=72.26999pt]{33p3qcase3b}\\\
(d)&(e)&(f)\\\ \end{array}$

Figure 8. Possible color assignments for a perfect precise 5-coloring of the
tiling $(3.3.p.3.q)$, with $p\neq q$.

The following proposition summarizes the preceding discussion.

######  Proposition 2.3.

Let ${\mathcal{T}}$ be the semiregular tiling $(3.3.p.3.q)$ with $p\neq q$.
Then the number of perfect precise $5$-colorings of ${\mathcal{T}}$ is

  1. (1)

one if and only if one of the following hold: one of $p$ and $q$ is a multiple
of $3$ but not of $4$ and the other is even that is not a multiple of $4$, one
of $p$ and $q$ is a multiple of $4$ but not of $3$ and the other is even that
is not a multiple of $4$, or one of $p$ and $q$ is a multiple of $4$ but not
of $3$ and the other is an odd multiple of $3$;

  2. (2)

two if and only if one of the following hold: one of $p$ and $q$ is a multiple
of $12$ and the other is even that is neither a multiple of $3$ nor of $4$,
both $p$ and $q$ are odd multiples of $3$, both $p$ and $q$ are multiples of
$4$ but not of $3$, or one of $p$ and $q$ is a multiple of $6$ but not of $4$
and the other is a multiple of $4$ but not of $3$;

  3. (3)

three if and only if one of the following hold: one of $p$ and $q$ is a
multiple of $6$ and the other is an odd multiple of $3$, or one of $p$ and $q$
is a multiple of $12$ and the other is a multiple of $4$ but not of $3$;

  4. (4)

four if and only if both $p$ and $q$ are multiples of $6$ but not of $4$;

  5. (5)

five if and only if one of $p$ and $q$ is a multiple of $12$ and the other is
a multiple of $6$ but not of $4$; and

  6. (6)

six if and only if both $p$ and $q$ are multiples of $12$.

The existence of three perfect precise 5-colorings for the tiling
$(3.3.3.3.6)$ in [2] agrees with Proposition 2.3. Examples of perfect precise
5-colorings of the tiling $(3.3.p.3.q)$ with $p\neq q$ are presented in Fig.
9.

$\begin{array}[]{ccc}\includegraphics[width=97.56493pt]{33334ppc1}&\includegraphics[width=101.17755pt]{33339ppc1}&\includegraphics[width=101.17755pt]{33339ppc2}\\\
(3.3.3.3.4)&(3.3.3.3.9)_{1}&(3.3.3.3.9)_{2}\\\
\includegraphics[width=101.17755pt]{3.3.6.3.12ppc1}&\includegraphics[width=101.17755pt]{3.3.6.3.12ppc2}&\includegraphics[width=101.17755pt]{3.3.6.3.12ppc4}\\\
(3.3.6.3.12)_{1}&(3.3.6.3.12)_{2}&(3.3.6.3.12)_{3}\end{array}$

$\begin{array}[]{cc}\includegraphics[width=101.17755pt]{3.3.6.3.12ppc3}&\includegraphics[width=101.17755pt]{3.3.6.3.12ppc5}\\\
(3.3.6.3.12)_{4}&(3.3.6.3.12)_{5}\end{array}$

Figure 9. Some perfect precise 5-colorings of the tiling $(3.3.p.3.q)$, with
$p\neq q$.

Recall that if $p=q$ then the tiling ${\mathcal{T}}$ has no perfect precise
$5$-coloring. Nevertheless, if we consider instead the action of the rotation
group $H=pp2$ on the tiling ${\mathcal{T}}$, then some of the discussion above
is applicable and results to chirally perfect precise 5-colorings of
${\mathcal{T}}$. We state this explicitly in the next proposition.

######  Proposition 2.4.

The number of chirally perfect precise $5$-colorings of the semiregular tiling
${{\mathcal{T}}=(3.3.p.3.p)}$ is

  1. (1)

one if and only if $p$ is an odd multiple of $3$ or $p$ is a multiple of $4$
but not of $3$;

  2. (2)

two if and only if $p$ is a multiple of $6$ but not of $4$; and

  3. (3)

three if and only if $p$ is a multiple of $12$.

Note that two chirally perfect precise colorings of the tiling $(3.3.4.3.4)$
using 5 colors were obtained in [2], each of which reflects into the other.
These two colorings correspond to the same partition of the tiling and in
effect, these colorings are the same. This is in agreement with Proposition
2.4. Fig. 10 shows the two chirally perfect precise 5-colorings of the tiling
$(3.3.6.3.6)$.

$\begin{array}[]{cc}\includegraphics[width=101.17755pt]{3.3.6.3.6a1}&\includegraphics[width=101.17755pt]{3.3.6.3.6b1}\\\
(3.3.6.3.6)_{1}&(3.3.6.3.6)_{2}\end{array}$

Figure 10. Some chirally perfect precise 5-colorings of the tiling
$(3.3.p.3.p)$.

###  2.4. 6-valent Semiregular Planar Tilings

We finish this section with a discussion on the perfect precise 6-colorings of
the family ${\mathcal{T}}=(3.p.3.q.3.q)$ of 6-valent semiregular tilings.

If $p=q=3$ then ${\mathcal{T}}=(3^{6})$ is the regular tessellation of the
Euclidean plane by equilateral triangles. Its unique perfect precise
$6$-coloring has been determined in [12]. If $p=q$ and $p>3$, then the
symmetry group of ${\mathcal{T}}=(3.p.3.p.3.p)$ is $G={\ast}p33$. Every vertex
of ${\mathcal{T}}$ is the center of a $3$-fold rotation. The stabilizer of
every $p$-gon (triangle) is a dihedral group $D_{p}$ ($D_{3}$). Two orbits of
tiles are formed under the action of $G$ on ${\mathcal{T}}$, namely the set of
$p$-gons and the set of triangles.

In Fig. 11, the six tiles meeting at vertex $v$ of ${\mathcal{T}}$ are
assigned different colors: the $p$-gon $t$ is colored $1$, and the succeeding
triangles and $p$-gons around $v$ are colored $2$, $3$, $4$, $5$, $6$, in that
order, such that triangle $t^{\prime}$ is colored $2$. Observe that
${\mathcal{T}}$ has a reflection symmetry $\alpha$ along the line that passes
through the center of $p$-gon $t$ and vertex $v$. For the $6$-coloring of
${\mathcal{T}}$ to be precise, the $p$-gon $a$ can only be assigned colors
$1$, $2$, or $6$. However, notice that $\alpha$ swaps colors $2$ and $6$. This
means that $p$-gon $a$ must get color $1$ so that the resulting coloring of
${\mathcal{T}}$ will be perfect.

Figure 11. Obtaining a perfect precise $6$-coloring of the tiling
$(3.p.3.p.3.p)$.

Let us denote by $\gamma$ the $3$-fold rotation about vertex $v$. Since
$\gamma$ is supposed to permute the colors in the coloring of ${\mathcal{T}}$,
we conclude that $\gamma$ maps colors $1$, $3$, and $5$ to $3$, $5$, and $1$,
respectively. This implies that the $p$-gon $b$ should be colored $5$.

We now consider the reflection symmetry $\beta$ of ${\mathcal{T}}$ along the
line that passes through the centers of $p$-gon $t$ and triangle $t^{\prime}$.
Notice that $\beta$ fixes colors $1$ and $2$, and swaps $3$ and $5$. This
implies that the $p$-gon $c$ should be colored $3$ and $p$-gon $d$ should get
color $1$. Now, $\beta$ either fixes or swaps colors $4$ and $6$. However,
triangles $e$ and $f$ are mapped to each other by the reflection $\beta$. For
the resulting $6$-coloring of ${\mathcal{T}}$ to be precise, we conclude that
$\beta$ swaps colors $4$ and $6$. Consequently, triangles $g$ and $h$ must be
respectively assigned colors $6$ and $4$. This means that the $p$ triangles
sharing an edge with $p$-gon $t$ should be colored $2$, $4$, and $6$ in a
cyclical manner. The coloring of ${\mathcal{T}}$ is now uniquely determined
and we see that $p$ must be a multiple of $3$ for a perfect precise
$6$-coloring of ${\mathcal{T}}$ to be obtained. The unique perfect precise
$6$-coloring of ${\mathcal{T}}$ generated corresponds to the partition
$O_{1}\cup O_{2}$ with $O_{1}=\left\\{g(J_{1}t_{1})\mid g\in G\right\\}$,
where $t_{1}$ is the $p$-gon stabilized by the $p$-fold rotation $QR$ and
$J_{1}=\langle PRQP,Q,R\rangle$ of index $3$ in $G$, and
$O_{2}=\left\\{g(J_{2}t_{2})\mid g\in G\right\\}$, where $t_{2}$ is the
triangle stabilized by the $3$-fold rotation $PR$ and $J_{2}=\langle
P,QRPRPQ,R\rangle$ of index $3$ in $G$. Thus we obtain the following
proposition.

######  Proposition 2.5.

The semiregular tiling $(3.p.3.p.3.p)$ has exactly one perfect precise
6-coloring if and only if $p$ is a multiple of $3$.

Fig. 12 shows the perfect precise $6$-colorings of the tilings $(3.6.3.6.3.6)$
and $(3.9.3.9.3.9)$.

$\begin{array}[]{cc}\includegraphics[width=101.17755pt]{3.6.3.6.3.6a}&\includegraphics[width=101.17755pt]{3.9.3.9.3.9a}\\\
(3.6.3.6.3.6)&(3.9.3.9.3.9)\\\ \end{array}$

Figure 12. Some perfect precise $6$-colorings of the tiling $(3.p.3.p.3.p)$.

If $p\neq q$, then the symmetry group of $\mathcal{T}=(3.p.3.q.3.q)$ is
$G=q{\ast}p$, a subgroup of index $2$ of the triangle group ${\ast}(2p)q2$
(where $P$ still denotes the reflection opposite angle $\pi/(2p)$). Three
orbits of tiles are formed under the action of $G$ on ${\mathcal{T}}$: the set
of $p$-gons, the set of $q$-gons, and the set of triangles. The stabilizer of
every $p$-gon ($q$-gon) in ${\mathcal{T}}$ is a dihedral group $D_{p}$ (cyclic
group $C_{q}$). We also note that no two $p$-gons are adjacent to each other,
two $q$-gons are attached to each vertex of every $p$-gon, and $q$ $q$-gons
are attached to every $q$-gon at the vertices.

As earlier, we start with six tiles meeting at vertex $v$ of ${\mathcal{T}}$
with the following color assignment: the $p$-gon $t$ is colored $1$, and the
succeeding triangles and $q$-gons around $v$ are colored $2$, $3$, $4$, $5$,
$6$, in that order, such that $q$-gon $t^{\prime}$ is colored $3$ (see Fig.
13). The tiling ${\mathcal{T}}$ still has the reflection symmetry $\alpha$
along the line that passes through the center of $p$-gon $t$ and vertex $v$.
Using similar arguments, we conclude that the $p$-gon $a$, and consequently
every $p$-gon in ${\mathcal{T}}$, should be colored $1$.

Figure 13. Obtaining a perfect precise 6-coloring of the tiling
$(3.p.3.q.3.q)$, with $p\neq q$.

For the resulting $6$-coloring of ${\mathcal{T}}$ to be precise, the $q$-gon
$b$ should be assigned colors $4$, $5$, or $6$. We consider these cases
separately, and we make use of the reflection symmetry $\beta$ of
${\mathcal{T}}$ along the line that passes through the centers of $p$-gon $t$
and the triangle colored $2$, and the $q$-fold rotation symmetry $\gamma$ of
${\mathcal{T}}$ about the center of $q$-gon $t^{\prime}$. Notice that $\beta$
fixes colors $1$ and $2$, and swaps color $3$ and the color of $q$-gon $b$,
while $\gamma$ fixes colors $1$ and $3$, and maps color $2$, $4$, $5$, and $6$
to the color of triangle $d$, color $2$, the color of $q$-gon $b$, and the
color of triangle $c$, respectively.

  1. (i)

Suppose $q$-gon $b$ is colored $5$. This implies that the $q$-gons around
$p$-gon $t$ are colored $3$ and $5$ alternately. Now, $\beta$ will either fix
or swap colors $4$ and $6$. However, triangles $c$ and $d$ should be colored
$4$ and $6$ and are mapped to each other by the reflection $\beta$. This means
that $\beta$ should swap colors $4$ and $6$, and so triangles $e$ and $f$
should be respectively colored $6$ and $4$. Hence, the $p$ triangles sharing
an edge with $p$-gon $t$ will be colored $2$, $4$, and $6$ in a cyclical
manner, which implies that $p$ is a multiple of $3$.

Let us go back to triangles $c$ and $d$. If triangle $c$ is colored $6$ and
triangle $d$ is colored $4$, then the $q$-fold rotation $\gamma$ effects the
color permutation $(2\,4)$. This implies that $q$ must be even (see Fig.
14(a)). On the other hand, if triangle $c$ is colored $4$ and triangle $d$ is
colored $6$, then $\gamma$ effects the color permutation $(2\,6\,4)$. Hence,
we have $q$ to be a multiple of $3$ (see Fig. 14(b)). In both cases, we obtain
a perfect precise $6$-coloring of ${\mathcal{T}}$ that corresponds to the
partition $O_{1}\cup O_{2}\cup O_{3}$, with $O_{1}$ the set of $p$-gons,
$O_{2}=\left\\{g(J_{2}t_{2})\mid g\in G\right\\}$, where $t_{2}$ is the
$q$-gon stabilized by the $q$-fold rotation $PR$ and $J_{2}=\langle
PR,QRQP\rangle$ of index $2$ in $G$, and $O_{3}=\left\\{g(J_{3}t_{3})\mid g\in
G\right\\}$, where $t_{3}$ is the triangle stabilized by the reflection $Q$.
If $q$ is even then $J_{3}=\langle Q,(PR)^{2},PR(QR)^{2}\rangle$ of index $3$
in $G$, and if $q$ is a multiple of $3$ then $J_{3}=\langle
Q,PRPQR,RPRQR\rangle$ of index $3$ in $G$.

  2. (ii)

Suppose $q$-gon $b$ is colored $4$. Then $\beta$ interchanges colors $3$ and
$4$ so triangle $e$ should be colored $3$, and interchanges colors $5$ and $6$
so triangle $f$ should be colored $5$. Therefore, the $p$ triangles sharing an
edge with $p$-gon $t$ will be colored $2$, $5$, $4$, $3$, and $6$ in a
cyclical manner. We obtain that $p$ must be a multiple of $5$.

We again have two different ways of assigning colors to triangles $c$ and $d$.
If triangle $c$ is colored $5$ and triangle $d$ is colored $6$, then the
$q$-fold rotation $\gamma$ induces the permutation $(2\,6\,5\,4)$ of colors.
Hence $q$ must be a multiple of $4$ (see Fig. 14(c)). If instead triangles $c$
and $d$ are colored $6$ and $5$, respectively, then $\gamma$ induces the
permutation $(2\,5\,4)$ of colors, and so $q$ is a multiple of $3$ (see Fig.
14(d)). For both possibilities, a perfect precise $6$-coloring of
${\mathcal{T}}$ is generated that corresponds to the partition $O_{1}\cup
O_{2}$, with $O_{1}$ the set of $p$-gons and $O_{2}=\left\\{g(Jt_{1}\cup
Jt_{2})\mid g\in G\right\\}$, where $t_{1}$ is the $q$-gon stabilized by the
$q$-fold rotation $PR$ and $t_{2}$ is the triangle stabilized by the
reflection $(RQ)^{3}R$. If $q$ is a multiple of $4$ then $J=\langle
PR,(RQ)^{3}R,QPR(PRQ)^{2},Q(PR)^{2}Q(RP)^{3}Q\rangle$ of index $5$ in $G$. On
the other hand, if $q$ is a multiple of $3$ then $J=\langle
PR,(RQ)^{3}R,Q(RP)^{3}Q,(QR)^{2}QPRQ\rangle$ of index $5$ in $G$.

  3. (iii)

Finally, suppose $q$-gon $b$ is colored $6$. Similar arguments show that the
$p$ triangles sharing an edge with $p$-gon $t$ will be colored $2$, $3$, $4$,
$5$, and $6$ in a cyclical manner, and so $p$ is again a multiple of $5$.

Assigning color $4$ and color $5$ respectively to triangles $c$ and $d$ will
mean that the $q$-fold rotation $\gamma$ effects the color permutation
$(2\,5\,6\,4)$. Hence $q$ must be a multiple of $4$ (see Fig. 14(e)). On the
other hand, if triangle $c$ is colored $5$ and triangle $d$ is colored $4$,
then $\gamma$ effects the color permutation $(2\,4)(5\,6)$ and so $q$ should
be even (see Fig. 14(f)). In both cases, we obtain a perfect precise
$6$-coloring of ${\mathcal{T}}$ that corresponds to the partition $O_{1}\cup
O_{2}$, with $O_{1}$ the set of $p$-gons and $O_{2}=\left\\{g(Jt_{1}\cup
Jt_{2})\mid g\in G\right\\}$, where $t_{1}$ is the $q$-gon stabilized by the
$q$-fold rotation $PR$ and $t_{2}$ is the triangle stabilized by the
reflection $(RQ)^{5}R$. If $q$ is a multiple of $4$ then $J=\langle
PR,(RQ)^{5}R,(PQR)^{2}Q,(QRPR)^{2}Q,Q(RP)^{2}(RQ)^{2}\rangle$ of index $5$ in
$G$. For the second possibility where $q$ is even, we have $J=\langle
PR,(RQ)^{5}R,(RQ)^{2}(PR)^{2}(QR)^{2},(QR)^{2}P(RQ)^{2}P\rangle$ of index $5$
in $G$.

$\begin{array}[]{ccc}\includegraphics[height=72.26999pt]{3p3q3qCase1a}&\includegraphics[height=72.26999pt]{3p3q3qCase1b}&\includegraphics[height=72.26999pt]{3p3q3qCase2a}\\\
(a)&(b)&(c)\\\
\includegraphics[height=72.26999pt]{3p3q3qCase2b}&\includegraphics[height=72.26999pt]{3p3q3qCase3a}&\includegraphics[height=72.26999pt]{3p3q3qCase3b}\\\
(d)&(e)&(f)\\\ \end{array}$

Figure 14. Possible color assignments for a perfect precise 6-coloring of the
tiling $(3.p.3.q.3.q)$, with $p\neq q$.

The following proposition summarizes the preceding discussion. We illustrate
some perfect precise $6$-colorings of the tiling $(3.p.3.q.3.q)$ with $p\neq
q$ for some values of $p$ and $q$ in Fig. 15.

$\begin{array}[]{ccc}\includegraphics[width=101.17755pt]{3.3.3.3.3.6ppc}&\includegraphics[width=101.17755pt]{3.4.3.4.3.3ppc}&\includegraphics[width=101.17755pt]{3.6.3.6.3.3ppc1}\\\
(3.6.3.3.3.3)&(3.3.3.4.3.4)&(3.3.3.6.3.6)_{1}\\\
\includegraphics[width=101.17755pt]{3.6.3.6.3.3ppc2}&\includegraphics[width=101.17755pt]{353636ppc1}&\includegraphics[width=101.17755pt]{353636ppc2}\\\
(3.3.3.6.3.6)_{2}&(3.5.3.6.3.6)_{1}&(3.5.3.6.3.6)_{2}\\\
\includegraphics[width=101.17755pt]{353434ppc1}&\includegraphics[width=101.17755pt]{353434ppc2}&\includegraphics[width=101.17755pt]{353434ppc3}\\\
(3.5.3.4.3.4)_{1}&(3.5.3.4.3.4)_{2}&(3.5.3.4.3.4)_{3}\end{array}$

Figure 15. Some perfect precise $6$-colorings of the tiling $(3.p.3.q.3.q)$,
with $p\neq q$.

######  Proposition 2.6.

Let ${\mathcal{T}}$ be the semiregular tiling $(3.p.3.q.3.q)$ with $p\neq q$.
Then the number of perfect precise $6$-colorings of ${\mathcal{T}}$ is

  1. (1)

one if and only if one of the following hold: $p$ is a multiple of $3$ but not
of $5$ and $q$ is even, $p$ is a multiple of $5$ but not of $3$ and $q$ is
even but not a multiple of $4$, $p$ is a multiple of $3$ but not of $5$ and
$q$ is an odd multiple of $3$, or $p$ is a multiple of $5$ but not of $3$ and
$q$ is an odd multiple of $3$;

  2. (2)

two if and only if one of the following hold: $p$ is a multiple of $15$ and
$q$ is even but not a multiple of $4$, $p$ is a multiple of $15$ and $q$ is an
odd multiple of $3$, $p$ is a multiple of $3$ but not of $5$ and $q$ is a
multiple of $6$, or $p$ is a multiple of $5$ but not of $3$ and $q$ is a
multiple of $6$ but not of $4$;

  3. (3)

three if and only if $p$ is a multiple of $5$ but not of $3$ and $q$ is a
multiple of $4$ but not of $3$;

  4. (4)

four if and only if one of the following hold: $p$ is a multiple of $15$ and
$q$ is a multiple of $4$ but not of $3$, $p$ is a multiple of $15$ and $q$ is
a multiple of $6$ but not of $4$, or $p$ is a multiple of $5$ but not of $3$
and $q$ is a multiple of $12$; and

  5. (5)

six if and only if $p$ is a multiple of $15$ and $q$ is a multiple of $12$.

##  3\. More Colors and Outlook

In Section 1, we saw that Theorem 1.6 may be used to generate all (chirally)
perfect precise colorings of a $k$-valent semiregular planar tiling
${\mathcal{T}}$. This process involves initially obtaining perfect colorings
of a tiling ${\mathcal{T}}$ and subsequently filtering out those that are not
precise. An advantage of this procedure is that it allows us to obtain perfect
colorings of semiregular planar tilings ${\mathcal{T}}$ where no two tiles
sharing a common vertex have the same color and the number of colors this time
is more than the valency of ${\mathcal{T}}$. For ease of reference, we shall
still refer to such colorings as perfect precise colorings.

To illustrate, we again consider the $(3.4.6.4)$ Euclidean tiling
${\mathcal{T}}$ with symmetry group $G$ of type $\ast 632$. Let us obtain a
perfect precise coloring of ${\mathcal{T}}$ with the least number of colors
such that the three $G$-orbits of ${\mathcal{T}}$, namely, the orbit $X_{1}$
of hexagons, the orbit $X_{2}$ of squares, and the orbit of $X_{3}$ of
triangles, do not share any colors. That is, we want a partition of
${\mathcal{T}}$ of the form $\cup_{i=1}^{3}\\{g(J_{i}t_{i})\mid g\in G\\}$ for
some suitable tile $t_{i}\in X_{i}$ and subgroup $J_{i}$ of $G$ such that
$J_{i}$ contains $\operatorname{Stab}_{G}(t_{i})$. Let us assign the same
color to each tile in the orbit $X_{1}$, and also the same color to each tile
in the orbit $X_{3}$. That is, we take $J_{1}=J_{3}=G$. For the coloring to be
precise, the orbit $X_{2}$ should be colored with at least three colors
because of the three squares that surround each triangle in ${\mathcal{T}}$.
This means that we need to find a subgroup $J_{2}$ of at least index $3$ in
$G$. We may take $J_{2}$ to be the subgroup $\langle PRQRP,Q,R\rangle$ of type
$\ast 632$ to obtain a coloring of the orbit $X_{2}$ with three colors that
correspond to the partition $\\{g(J_{2}t_{2})\mid g\in G\\}$ where $t_{2}$ is
the square stabilized by the two-fold rotation $PRQRPR$. This will yield a
perfect precise coloring of ${\mathcal{T}}$ with five colors. Another
possibility is to take $J_{2}$ to be the subgroup $\langle P,Q,QRPQRQ\rangle$
of type $2{\ast}22$ where $t_{2}$ is the square stabilized by the two-fold
rotation $PQ$. The perfect precise $5$-colorings of ${\mathcal{T}}$ obtained
are shown in Fig. 16.

$\begin{array}[]{cc}\includegraphics[width=144.54pt]{5colors3464a}&\includegraphics[width=144.54pt]{5colors3464}\\\
\text{subgroup }J_{2}\text{ of type}\ast 632&\text{subgroup }J_{2}\text{ of
type }2*22\end{array}$

Figure 16. Perfect precise 5-colorings of the tiling $(3.4.6.4)$.

We also provide examples of perfect precise colorings (with more than $4$
colors) of the $(6.4.6.4)$ hyperbolic tiling ${\mathcal{T}}$. The orbit
$X_{1}$ of hexagons and the orbit $X_{2}$ of squares are the two orbits of
${\mathcal{T}}$ formed under the action of its symmetry group $G$ of type
$\ast 642$. Consider the following subgroups of $G$: $J_{1}=\langle
PRP,Q,R\rangle$ of index $2$, $J_{2}=\langle P,R,QRPRQ,QRQRQ\rangle$ of index
$3$, ${K_{1}=\langle PQRQRP,Q,R\rangle}$ of index $4$, and $K_{2}=\langle
P,R,QRQ\rangle$ of index $2$. Take $t_{1}$ to be the hexagon stabilized by the
six-fold rotation $QR$ and $t_{2}$ to be the square stabilized by the two-fold
rotation $PR$. Then the $5$-coloring corresponding to the partition
$\cup_{i=1}^{2}\left\\{g(J_{i}t_{i})\mid g\in G\right\\}$ of ${\mathcal{T}}$
and the $6$-coloring corresponding to the partition
$\cup_{i=1}^{2}\left\\{g(K_{i}t_{i})\mid g\in G\right\\}$ of ${\mathcal{T}}$
are perfect and precise. The colorings are shown in Fig. 17.

$\begin{array}[]{cc}\includegraphics[width=101.17755pt]{6464ppc5colors}&\includegraphics[width=101.17755pt]{6464ppc6colors}\end{array}$

Figure 17. Perfect precise 5-coloring and $6$-coloring of the tiling
$(6.4.6.4)$.

In this contribution, we have presented two approaches to obtain perfect
precise colorings of semiregular planar tilings. The first method involves a
combinatorial analysis of the possible ways to color the tiling such that the
resulting coloring is precise and perfect. By applying this method, we have
identified all perfect precise colorings for certain families of $k$-valent
semiregular planar tilings where $k\leq 6$. Alternatively, group-theoretic
methods can also be used to obtain perfect precise colorings of tilings. This
approach is applicable in finding the perfect precise coloring of a specific
semiregular tiling. It is also particularly useful when dealing with a broader
definition of perfect precise colorings, wherein the number of colors used
exceeds the valency of the tiling.

Recall that in a semiregular planar tiling ${\mathcal{T}}$, the action of the
symmetry group on the set of vertices is transitive. As a result, when we have
a perfect coloring of ${\mathcal{T}}$, we only need to ensure that no two
tiles sharing one vertex have the same color to confirm that the coloring is
also precise. Building on this observation, it would be interesting to
determine the perfect precise colorings of isogonal tilings, as well as
$k$-isogonal tilings. In Fig. 18, we provide an example of a perfect precise
coloring of the isogonal tiling IG 71.

Figure 18. A perfect precise coloring of the isogonal tiling IG 71.

## Acknowledgements

This article is dedicated to the memory of our mentor, Professor René P.
Felix. The authors are grateful to the anonymous referees for their valuable
remarks, which have helped improve the quality of the paper.

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