further $2i$ pairs. Indeed, each new diamond $abx,aby$ $2$-claims a pair $xy$
$1$-claimed by some previous edge $xyz$, and since $i\geqslant 3$, the pairs
$za$ and $zb$ are $\\{3,5\\}$-claimed by the final $2$-cluster $F$. Thus, we
have

$\displaystyle w(F)$ $\displaystyle\geqslant$
$\displaystyle(5i+3)\,h(1,3)+2i\cdot h(3,5)\ =\ (5i+3)+2i\cdot\frac{11}{61}$
$\displaystyle=$ $\displaystyle\frac{18-3i}{61}+\frac{165}{61}\cdot(2i+1)\
\geqslant\ \frac{165}{61}\cdot|F|.$

Thus, suppose that $|F|\leqslant 6$. By $(8,6)$-freenees, we have that
$|F|\leqslant 5$. First, consider the case that $F=F_{1}\cup F_{2}$ for
$(2)$-mergeable $F_{1},F_{2}\in\mathcal{M}_{1}$ via some pair $uv$ (thus $1\in
C_{F_{1}}(uv)\setminus C_{F_{2}}(uv)$). Let $F_{1}$ be an $i$-tree and $F_{2}$
be a $j$-tree. Then, $F$ has $i+j$ edges and $1$-claims $2i+2j+2$ pairs. Note
that $j\geqslant 2$ as $F_{2}$ has to contain a diamond. Also, since $G$ is
${\mathcal{G}}^{(3)}_{6}$-free, the subgraphs $F_{1}$ and $F_{2}$ do not share
any further vertices in addition to $u$ and $v$.

Suppose first that $i=1$. Every pair in $P_{1}(F)$ is $\\{1,3\\}$-claimed by
$F$. If $j=2$, then $F$ $3$-claims the remaining $2$ pairs and we have

$w(F)=8\,h(1,3)+2\,h(3)=8+2\cdot\frac{6}{61}=\frac{500}{61}>\frac{165}{61}\cdot
3.$

If $j\geqslant 3$, then $F$ $\\{3,4\\}$-claims at least $2$ pairs not in
$P_{1}(F)$. Hence,

$w(F)\geqslant(2j+4)\,h(1,3)+2\,h(3,4)=(2j+4)+2=\frac{(201-43j)}{61}+\frac{165}{61}\,(j+1),$

which is at least $\frac{165}{61}\,(j+1)$ since $j=|F|-1\leqslant 4$, as
desired.

Suppose that $i\geqslant 2$. Then, every pair in $P_{1}(F)$ is
$\\{1,2\\}$-claimed by $F$. Also, given an edge $uvx$ in $F_{1}$ and a diamond
$abu,abv$ in $F_{2}$, the $2$-cluster $F$ $\\{3,4\\}$-claims the pairs
$ax,bx\notin P_{1}(F)$. Moreover, if $i=2$, then the (unique) pair in
${V(F_{1})\choose 2}\setminus P_{1}(F_{1})$ is $\\{2,4\\}$-claimed by $F$;
combining this with the fact that $j=|F|-i\leqslant 3$ yields

$\displaystyle w(F)$ $\displaystyle\geqslant$
$\displaystyle(2j+6)\,h(1,2)+2\,h(3,4)+h(2,4)\ =\ (2j+6)+2+\frac{1}{2}$
$\displaystyle=$ $\displaystyle\ \frac{377-86j}{122}+\frac{165}{61}\,(j+2)>\
\frac{165}{61}\,|F|.$

If $i=3$, then $j=2$. Again, $F$ $\\{1,2\\}$-claims all 12 pairs in $P_{1}(F)$
and $\\{3,4\\}$-claims another 2 pairs, but it also $\\{2,3\\}$-claims at
least $2$ other pairs, namely the pairs 2-claimed but not $1$-claimed by
$F_{1}$. Hence, we have

$w(F)\geqslant
12\,h(1,2)+2\,h(3,4)+2\,h(2,3)=12+2+2\cdot\frac{36}{61}=\frac{926}{61}>\frac{165}{61}\cdot
5.$

Now, note that a $2$-cluster made of at least four $1$-clusters has at least 6
edges. Thus, it remains to consider the case when $F$ is obtained by merging
three trees $F_{1},F_{2},F_{3}\in\mathcal{M}_{1}$. We cannot have
$|F|\leqslant 4$ as then, the $2$-cluster $F$ would be made of at least two
1-trees and at most one 2-tree, which is impossible for $3$-graphs. Hence, we
obtain that $|F|=5$ with the composition $(3,1,1)$ or $(2,2,1)$.

If $F$ has composition $(3,1,1)$, then $F$ $\\{1,3\\}$-claims all $13$ pairs
in $P_{1}(F)$ and $\\{3,4\\}$-claims at least $4$ other pairs (namely, for
each 1-tree $xyz$ and the corresponding diamond $aby,abz$ inside the 3-tree,
the pairs $ax$ and $bx$ are $\\{3,4\\}$-claimed by $F$). We have that

$w(F)\geqslant 13\,h(1,3)+4\,h(3,4)=13+4=\frac{1037}{61}>\frac{165}{61}\cdot
5.$

Suppose that $F$ has composition $(2,2,1)$. Then, $F$ $\\{1,3\\}$-claims all
$13$ pairs in $P_{1}(F)$. Also, $F$ can be built from its $1$-tree by
attaching each new diamond $abx,aby$ via a previously $1$-claimed pair $xy$,
say by $xyz\in F$; here each of the pairs $az$ and $bz$ is $\\{3,5\\}$-claimed
by the final $2$-cluster $F$. Thus, we have that

$w(F)\geqslant
13\,h(1,3)+4\,h(3,5)=13+4\cdot\frac{11}{61}=\frac{837}{61}>\frac{165}{61}\cdot
5,$

which finishes the case analysis and the proof. ∎

Finally, we prove the following claim.

######  Claim 5.16.

For all $F\in\mathcal{M}_{3^{+}}\setminus\mathcal{M}_{2}$, we have that
$w(F)\geqslant\frac{165}{61}\,|F|$.

###### Proof.

Recall that if $F_{1},F_{2}\in\mathcal{M}_{2}$ are $({3^{+}})$-mergeable via
$uv$, then $1\in C_{F_{1}}(uv)$ and $3\in C_{F_{2}}(uv)$ and, in addition,
either $2\in C_{F_{1}}(uv)$ or $4\in C_{F_{2}}(uv)$. In particular, we have
that $1,2\notin C_{F_{2}}(uv)$ (as otherwise we would have already merged
$F_{1}$ and $F_{2}$ when constructing $\mathcal{M}_{2}$); also, by (24)
$5\not\in C_{F_{2}}(uv)$ and $3\not\in C_{F_{1}}(uv)$.

Let $F\in\mathcal{M}_{3^{+}}\setminus\mathcal{M}_{2}$ be made of
$F_{1},\dots,F_{s}\in\mathcal{M}_{2}$ ${3^{+}}$-merged in this order as in
Claim 5.12. Assume without loss of generality that $F_{1}$ and $F_{2}$ are
$(3^{+})$-mergeable via some pair $uv$. Let $F^{\prime}:=F_{1}\cup F_{2}$. As
$1,2\notin C_{F_{2}}(uv)$ and $3\in C_{F_{2}}(uv)$, we know that there is a
$(5,3)$-configuration in $F_{2}$ containing $uv$ which is either a $3$-path
$P_{3}=\\{uab,abc,bcv\\}\subseteq F_{2}$ or a $2$-cluster
$C_{3}=\\{aub,auc,bcv\\}\subseteq F_{2}$ (which is the union of a $1$-tree and
a $2$-tree that are $(2)$-mergeable), see Figure 3.

Figure 3: Configurations $P_{3}$ and $C_{3}$.

Since $3\notin C_{F_{1}}(uv)$ by (24), $F_{1}$ cannot be an $i$-tree for
$i\geqslant 3$. We split the proof into $3$ cases depending on whether $F_{1}$
is a $2$-tree, a $1$-tree or an element of
$\mathcal{M}_{2}\setminus\mathcal{M}_{1}$.

Case 1. Assume that $F_{1}$ is a $2$-tree.

Then, $1,2\in C_{F_{1}}(uv)$. By (24), we have $4,5\notin C_{F_{2}}(uv)$, and
recall that $F_{2}$ contains one of $P_{3}$ or $C_{3}$ as described above.
Also, the union $F^{\prime\prime}:=F_{1}\cup P_{3}$ or $F_{1}\cup C_{3}$ is a
$(7,5)$-configuration, so

$\binom{V(F^{\prime\prime})}{2}\cap P_{1}(G\setminus
F^{\prime\prime})=\emptyset.$ (27)

Moreover, since $5\notin C_{F_{2}}(uv)$, no pair in $V(P_{3})$ (resp.
$V(C_{3})$) can be $2$-claimed by $G\setminus F^{\prime}$. It follows that the
$2$-cluster $F_{2}$ is equal to $P_{3}$ or $C_{3}$, and thus
$F^{\prime\prime}=F^{\prime}$.

Suppose that $F$ consists of $s\geqslant 3$ 2-clusters. Let the next
${3^{+}}$-merging step (of $F^{\prime}$ and $F_{3}$) be via a pair
$u^{\prime}v^{\prime}$. It follows from (27) (and from
$F^{\prime\prime}=F^{\prime}$) that $u^{\prime}v^{\prime}\in
P_{1}(F^{\prime})\cap P_{3}(F_{3})$. By (24), it holds that $3\notin
C_{F^{\prime}}(u^{\prime}v^{\prime})$. As $F_{2}$ contains 3 edges,
necessarily $u^{\prime}v^{\prime}\in P_{1}(F_{1})$. Also, since
$u^{\prime}v^{\prime}$ is in $P_{1}(F_{1})\subseteq P_{2}(F_{1})$, we see that
$F_{1}$ alone and $F_{3}$ are $({3^{+}})$-mergeable via
$u^{\prime}v^{\prime}$. Now, our previous argument about the structure of
$F_{2}$ applies to $F_{3}$ as well and shows that $F_{3}$ is isomorphic to a
copy of $P_{3}$ or $C_{3}$ 3-claiming $u^{\prime}v^{\prime}$. Furthermore, the
$(5,3)$-configurations $F_{2}$ and $F_{3}$ can share at most one vertex, so
${V(F_{2})\choose 2}\cap{V(F_{3})\choose 2}=\emptyset$. The very same
reasoning applies in turn to each $F_{i}$ with $i\in[4,s]$ with the analogous
conclusion. Indeed, by Lemma 5.11, there is $j\in[1,i-1]$ such that $F_{i}$
and $F_{j}$ ${3^{+}}$-merge. Together with the $(8,6)$-freeness and the fact
that $F_{1}\cup F_{2},\hskip 0.9pt.\hskip 0.3pt.\hskip 0.3pt.\hskip
1.5pt,F_{1}\cup F_{i-1}$ form $(7,5)$-configurations, $F_{i}$ can only
${3^{+}}$-merge with $F_{1}$. Let us bound from below the total weight
assigned by $F$, being rather loose in our estimates. The five pairs in
$P_{1}(F_{1})$ are $\\{1,2\\}$-claimed (and we just ignore the remaining pair
inside $V(F_{1})$ which is $2$-claimed). For each $i\in[2,s]$, if $F_{i}$ is a
3-path, then it $\\{1,3\\}$-claims all seven pairs in its $2$-shadow
$P_{1}(F_{i})$ and two further pairs inside $V(F_{i})$ are
$\\{2,3,4\\}$-claimed by $F_{1}\cup F_{i}\subseteq F$ (for example, if $i=2$,
then these are the pairs $uc$ and $av$). All these pairs are unique to $F_{i}$
and thus $F_{i}$ contributes at least $7\,h(1,3)+2\,h(2,3,4)=9$ to $w(F)$.
Also, if $F_{i}$ is isomorphic to $C_{3}$, then it $\\{1,3\\}$-claims all 8
pairs in $P_{1}(F_{i})$ and one further pair inside $V(F_{i})$ is
$\\{3,4\\}$-claimed by $F_{1}\cup F_{i}$ (for example, if $i=2$ then it is the
pair $av$). Thus, $F_{i}$ contributes at least $8\,h(1,3)+h(3,4)=9$ to $w(F)$.
We conclude by $s\geqslant 2$ (which follows from $F\notin\mathcal{M}_{2}$)
that

$\displaystyle w(F)$ $\displaystyle\geqslant$ $\displaystyle 5\,h(1,2)+9(s-1)\
=\ 5+9(s-1)$ $\displaystyle=$
$\displaystyle\frac{54s-79}{61}+\frac{165}{61}\cdot(3s-1)\ >\
\frac{165}{61}\,|F|,$

as desired.

Case 2. Assume that $F_{1}$ is a $1$-tree $\\{uvw\\}$.

As $F_{1}$ and $F_{2}$ are $({3^{+}})$-mergeable via $uv$, we have $3,4\in
C_{F_{2}}(uv)$. Thus, $|F_{2}|\geqslant 4$ and $|F_{2}|\notin\\{5,6\\}$
(otherwise, we would get a $(8,6)$-configuration). Also, $1,2\notin
C_{F_{2}}(uv)$ since $F_{1},F_{2}\in\mathcal{M}_{2}$ are distinct. Hence, as
before, $F_{2}$ contains a copy of $P_{3}$ or $C_{3}$ 3-claiming $uv$.

Suppose first that $|F_{2}|\geqslant 7$. Then, $F_{2}$ has the structure given
by Lemma 5.9a, consisting of a 1-tree or 3-tree with a number of diamonds
merged in one by one. A $(5,3)$-configuration in $F_{2}$ that contains $uv$
involves at most two $1$-clusters of $F_{2}$, whose union $F^{\prime\prime}$
has at most $5$ edges; moreover, if $F^{\prime\prime}$ consists of two
$1$-clusters, then these $1$-clusters are $2$-mergeable. The proof of Lemma
5.9a shows that, if we build $F_{2}\in\mathcal{M}_{2}$ by starting with the
partial $2$-cluster $F^{\prime\prime}$ and attaching $1$-clusters one by one,
then we reach a $(7,5)$-configuration $F^{\prime\prime\prime}$ just before the
number of edges jumps over $6$. However, $F_{1}$ shares at least $2$ vertices
with $F^{\prime\prime\prime}\supseteq F^{\prime\prime}$, so $F_{1}\cup
F^{\prime\prime\prime}$ is a $(8,6)$-configuration. This contradiction shows
that $|F_{2}|=4$.

Now, denote the $4$-edge hypergraph $F_{2}$ by $T_{4}$ if $F_{2}$ consists of
(a copy of) $P_{3}$ that 3-claims $uv$ together with one more edge (thus,
$T_{4}$ is a $4$-tree or a $2$-cluster made of a $3$-tree and a $1$-tree as
its $1$-clusters), and denote $F_{2}$ by $C_{4}$ if $F_{2}$ consists of (a
copy of) $C_{3}$ that 3-claims $uv$ with one more edge (thus, the $1$-clusters
of $C_{4}$ are two $2$-trees, or a $3$-tree and a $1$-tree). Recall that
$F^{\prime}=F_{1}\cup F_{2}$.

Suppose first that $s\geqslant 3$. Let $F_{3}$ and $F^{\prime}$ be merged via
a pair $u^{\prime}v^{\prime}$. The $3$-graphs $F_{3}$ and $F^{\prime}$ cannot
be $({3^{+}})$-mergeable via $u^{\prime}v^{\prime}$ as otherwise an edge in
$F_{3}$ containing $u^{\prime}v^{\prime}$ together with $F^{\prime}$ would be
a $(8,6)$-configuration. So $F^{\prime}$ and $F_{3}$ are
$({3^{+}})$-mergeable, that is, $1\in C_{F^{\prime}}(u^{\prime}v^{\prime})$
and $3\in C_{F_{3}}(u^{\prime}v^{\prime})$. Then, $3\notin
C_{F^{\prime}}(u^{\prime}v^{\prime})$. This greatly limits the number of
possibilities for the pair $u^{\prime}v^{\prime}$ inside $P_{1}(F^{\prime})$.
If $F_{2}=T_{4}$, one can easily notice that $u^{\prime}v^{\prime}$ cannot be
in $P_{1}(F_{2})\cup\\{uv\\}$ and thus $u^{\prime}v^{\prime}\in
P_{1}(F_{1})\setminus\\{uv\\}$. Hence, $F_{1}$ alone and $F_{3}$ are
$({3^{+}})$-mergeable and the above analysis for $F_{2}$ applies to $F_{3}$ as
well, showing that the $2$-cluster $F_{3}$ is isomorphic to a copy of $T_{4}$
or $C_{4}$ 3-claiming $u^{\prime}v^{\prime}$. If $F_{2}=C_{4}$, then either
$u^{\prime}v^{\prime}\in P_{1}(F_{1})\setminus\\{uv\\}$ (and, again, $F_{3}$
is isomorphic to $T_{4}$ or $C_{4}$) or, for some vertices $a,b,c,d$, we have
$F_{2}=\\{aub,auc,bcv,cdv\\}$ and $u^{\prime}v^{\prime}$ is $cd$ or $dv$;
also, the argument of Case $1$ (with $\\{bcv,cdv\\}$ playing the role of the
2-tree $F_{1}$ from Case $1$) shows that $F_{3}$ rooted at
$u^{\prime}v^{\prime}$ is isomorphic to $P_{3}$ or $C_{3}$, no other $F_{i}$
with $i\geqslant 4$ can be ${3^{+}}$-merged with $F_{3}$, and all pairs in
${V(F_{3})\choose 2}$ are unique to $F_{3}$. Using Lemma 5.11, the same
argument applies to each new $F_{i}$ with $i\geqslant 4$. To summarise, we
obtained that each $F_{i}$ for $i\geqslant 2$ is either some instance of
$T_{4}$ or $C_{4}$ ${3^{+}}$-merged with the single edge $F_{1}$, or an
instance of $P_{3}$ or $C_{3}$ ${3^{+}}$-merged with a copy of $C_{4}$ as
specified above; also, the only pairs shared between these $2$-clusters are
the pairs along which these ${3^{+}}$-mergings occur.

Now, assume the final $F$ consists of one $1$-tree, $i$ copies $T_{4}$ or
$C_{4}$, and $j$ copies of $P_{3}$ or $C_{3}$. (Although we could say more
about the structure of $F$, e.g. that $i\leqslant 3$ and $j\leqslant 2i$,
these observations are not needed for our estimates.) Then, $|F|=1+4i+3j$.
Each copy of $T_{4}$ (which is a 4-tree or a 3-tree ${2}$-merged with a single
edge) has, in addition to the pair via which it is merged with $F_{1}$, at
least nine $\\{1,3\\}$-claimed pairs and other two $\\{2,3,4\\}$-claimed
pairs. Thus, it contributes at least $9\,h(1,3)+2\,h(2,3,4)=11$ to $w(F)$.
Likewise, each copy of $C_{4}$ contributes at least
$8\,h(1,3)+2\,h(1,2)+h(3,4)=11$ to $w(F)$. Also, as in Case 1, each copy of
$P_{3}$ or $C_{3}$ contributes at least $9$ to $w(F)$. Additionally, we have
$3$ pairs in $P_{1}(F_{1})$ which are $\\{1,4\\}$-claimed by $F_{1}\cup
F_{2}$. Hence, we get the required:

$\displaystyle w(F)$ $\displaystyle\geqslant$ $\displaystyle 3\,h(1,4)+11i+9j\
=\ 3\cdot\frac{55}{61}+11i+9j$ $\displaystyle=$
$\displaystyle\frac{11i+54j}{61}+\frac{165}{61}\,(1+4i+3j)\ >\
\frac{165}{61}\,|F|.$

Case 3. Assume that $F_{1}\in\mathcal{M}_{2}\setminus\mathcal{M}_{1}$.

Here, $F_{1}$ is a $2$-merging of at least two $1$-clusters and thus has at
least 3 edges. Let $T_{1}\in\mathcal{M}_{1}$ be the $1$-cluster in $F_{1}$
1-claiming $uv$ (recall that $uv$ is the pair via which $F_{1}$ and $F_{2}$
are $({3^{+}})$-mergeable). The tree $T_{1}$ has at most 2 edges as otherwise
$T_{1}$ together with $F_{2}$ would contain an $(8,6)$-configuration, a
contradiction. Also, $T_{1}$ cannot be a $1$-tree as otherwise $F_{1}\setminus
T_{1}$ would contain a diamond $2$-claiming a pair in $P_{1}(T_{1})$ (since
$F_{1}\in\mathcal{M}_{2}\setminus\mathcal{M}_{1}$), which implies that $3\in
C_{F_{1}}(uv)$, a contradiction. Therefore, $T_{1}$ is a $2$-tree. As
$F_{1}\in\mathcal{M}_{2}\setminus\mathcal{M}_{1}$, $T_{1}$ has to be
$2$-merged with some other $1$-cluster $T_{2}\subseteq F_{1}$. It is
impossible that $T_{2}$ and $T_{1}$ are $(2)$-mergeable as otherwise $T_{1}$
plus an edge of $T_{2}$ would be a $(5,3)$-configuration containing $uv$ in
$F_{1}$, which contradicts $3\notin C_{F_{1}}(uv)$. Thus, $T_{1}$ and $T_{2}$
are $(2)$-mergeable. Also, $T_{2}$ has at most 3 edges since trees with more
edges would form an $(8,6)$-configuration with $T_{1}$. It is routine to check
that we can assume (after swapping $u$ and $v$ if necessary) that, for some
vertices $a,b\in V$, the 2-tree $T_{1}$ is $\\{avu,avb\\}$ and the pair
$2$-claimed by $T_{2}$ is $ab$ or $bv$. Also, any further 2-merging involving
$T_{1}\cup T_{2}$ would cause an $(8,6)$-configuration. We conclude that
$F_{1}=T_{1}\cup T_{2}$.

The same argument as in Case 1 shows that $F_{2}$ is given by $P_{3}$ (a
3-path) or $C_{3}$ (a diamond and a single edge), see Figure 3. Let
$F^{\prime}:=F_{1}\cup F_{2}$. We have $2$ subcases depending on $T_{2}$.

Case 3-1. Assume that $|T_{2}|=2$.

Suppose first that $s\geqslant 3$. If $F_{3}$ and $F^{\prime}$ are
$({3^{+}})$-mergeable via some pair $u^{\prime}v^{\prime}$ then, by
$(8,6)$-freeness, $u^{\prime}v^{\prime}$ must be one of the two pairs
$\overline{1}3$-claimed by $F_{1}$, and $F_{3}$ is a $1$-tree; therefore, by
Claim 5.12, we can reorder $2$-clusters constituting $F$ starting with $F_{3}$
and follow the analysis in the Case 2. Now assume that both pairs
$\overline{1}3$-claimed by $F_{1}$ are not used for further
${3^{+}}$-mergings. Thus, $F^{\prime}$ and $F_{3}$ are $({3^{+}})$-mergeable
via $u^{\prime}v^{\prime}$. Then, since $3\in
C_{F_{3}}(u^{\prime}v^{\prime})$, $u^{\prime}v^{\prime}$ must be in
$P_{1}(F^{\prime})\setminus P_{3}(F^{\prime})=\\{au\\}$ (recall that $3\in
C_{F_{2}}(uv)\subseteq C_{F^{\prime}}(uv)$), $F_{3}$ rooted at
$u^{\prime}v^{\prime}=au$ is isomorphic to $P_{3}$ or $C_{3}$; moreover no
further ${3^{+}}$-mergings are possible and thus $s=3$. Thus, for both $s=2$
and $s=3$, we can assume the final $3^{+}$-cluster $F$ is made of $F_{1}$ and
$j$ copies of $P_{3}$ or $C_{3}$ where $j\in\\{1,2\\}$. Then, $|F|=4+3j$,
$F_{1}$ contributes at least
$8\,h(1,3)+2\,h(1,2)+2h(3,4)+h(2,4)=10+2+\frac{1}{2}=\frac{25}{2}$ to $w(F)$.
Hence, we have

$\displaystyle w(F)$ $\displaystyle\geqslant$ $\displaystyle\frac{25}{2}+9j\
=\ \frac{108j+205}{122}+\frac{165}{61}\,(4+3j)\ >\ \frac{165}{61}\,|F|,$

as desired.

Case 3-2. Assume that $T_{2}$ is a 3-tree.

Here $C_{F_{1}}(uv)\supseteq\\{1,2,4,5\\}$ so $F_{2}$ has exactly 3 edges and
$F^{\prime}=F_{1}\cup F_{2}$ is a $(10,8)$-configuration. Suppose first that
$s\geqslant 3$. Since no $3$-claimed pair of $F^{\prime}$ can be $1$-claimed
by $G\setminus F^{\prime}$, the $3$-graphs $F_{3}$ and $F^{\prime}$ cannot be
$({3^{+}})$-mergeable. So let $F^{\prime}$ and $F_{3}$ be
$({3^{+}})$-mergeable via some pair $u^{\prime}v^{\prime}$. Since $3\in
C_{F_{3}}(u^{\prime}v^{\prime})$, $u^{\prime}v^{\prime}$ must be in
$P_{1}(F^{\prime})\setminus P_{3}(F^{\prime})=\\{au\\}$; furthermore, $F_{3}$
rooted at $u^{\prime}v^{\prime}=au$ is isomorphic to $P_{3}$ or $C_{3}$, no
further ${3^{+}}$-mergings are possible and $s=3$. Thus, for both $s=2$ and
$s=3$, the final $3^{+}$-cluster $F$ consists of $F_{1}$ and $j$ copies of
$P_{3}$ or $C_{3}$ where $j\in\\{1,2\\}$. Here, $|F|=5+3j$. Note that $F_{1}$
contributes at least $10\,h(1,3)+2\,h(1,2)+4\,h(3,4)=12+4=16$ to the total
weight. We have

$w(F)\ \geqslant\ 16+9j\ =\ \frac{54j+151}{61}+\frac{165}{61}\,(5+3j)\ >\
\frac{165}{61}\,|F|.$

This finishes the proof of the claim. ∎

Hence, by the previous claims, we conclude that

$|G|=\sum_{F\in\mathcal{M}_{3^{+}}}|F|\leqslant\frac{61}{165}\sum_{F\in\mathcal{M}_{3^{+}}}w(F)=\frac{61}{165}\sum_{uv\in\binom{V(H)}{2}}w(uv)\leqslant\frac{61}{165}\binom{n}{2}.$

This proves Theorem 1.3. ∎

##  6 Concluding remarks

In this paper, we made progress on the Brown–Erdős–Sós Problem (the case of
3-uniform hypergraphs) with $k\in\\{5,6,7\\}$ edges and its extension to
$r$-graphs. We note that a further extension of the Brown–Erdős–Sós Problem
proposed by Shangguan and Tamo [ShTamo] asks to determine whether the limits

$\lim_{n\rightarrow\infty}n^{-t}f^{(r)}(n;k(r-t)+t,k)$

exist for all fixed $r,t,k$ and, if so, to find their values. (The case that
we studied here corresponds to $t=2$.)

Let us briefly summarise what is known. The results in [BES, Rodl85] resolve
the case $k=2$. In [G, GJKKLP, ShTamo], this problem was completely solved
when $k\in\\{3,4\\}$. In [DP, sh], the existence of the limit was proved for
$t=2$. In [letzter2023problem], the value of the limit (and thus its
existence) was established for even $k$ when $r\gg k,t$ is sufficiently large.
Also, it was proved in [letzter2023problem] that if $k\in\\{5,7\\}$ then then
limit exists for any $r$ and $t$. Our results determine the limit values when
$t=2$ and $k\in\\{5,6,7\\}$.

It would be interesting to study the existence of limits for the remaining
sets of parameters as well as their precise values.

## Acknowledgments

We thank Felix Joos and Marcus Kühn for useful discussions. In particular,
some starting ideas of the present paper originated from the joint work
[GJKKLP].

## References


# Elastocaloric Effect in Graphene Kirigami

Luiz A. Ribeiro Junior1,†, Marcelo L. Pereira Junior2,‡, and Alexandre F.
Fonseca3,§

1University of Brasília, Institute of Physics, Brasília, Brazil
2University of Brasília, Faculty of Technology, Department of Electrical
Engineering, Brasília, Brazil
3Applied Physics Department, Gleb Wataghin Institute of Physics, University of
Campinas, Campinas, São Paulo, Brazil.
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>.

Abstract: Kirigami, a traditional Japanese art of paper-cutting, has recently
been explored for its elastocaloric effect (ECE) in kirigami-based materials
(KMs), where applying strain induces temperature changes. In this study, we
investigate the ECE in a nanoscale graphene kirigami (GK) monolayer,
representing the thinnest possible KM, to better understand this phenomenon.
Through molecular dynamics simulations, we analyze the temperature change and
coefficient of performance (COP) of the nanoscale GK architecture. Our
findings reveal that while GKs lack the intricate temperature changes observed
in macroscopic KMs, they exhibit a substantial temperature change of
approximately 9.32 K (23 times higher than that of macroscopic KMs, which is
about 0.4K) for heating and -3.50 K for cooling. Additionally, they
demonstrate reasonable COP values of approximately 1.57 and 0.62,
respectively. It is noteworthy that the one-atom-thick graphene configuration
prevents the occurrence of the complex temperature distribution observed in
macroscopic KMs.

##  1 Introduction

Graphene is a two-dimensional allotrope of carbon atoms arranged in a
hexagonal lattice [1]. It was first isolated in 2004 using the mechanical
exfoliation technique [2]. Graphene’s exceptional combination of mechanical
[3], electronic [4], thermal [5, 6] and optical properties [7] has made it one
of the most extensively studied systems in materials science, with vast
potential for use in various applications such as flat electronics [8], energy
storage [9], and biomedical engineering [10]. However, incorporating graphene
into flexible devices poses challenges because of its intrinsically high
stiffness, which makes it difficult to bend or stretch. In addition, graphene
is brittle, with only a few percent tensile fracture strains [11].

Graphene Kirigami (GK) is a cutting-edge technique miming the ancient Japanese
art of paper-cutting to manipulate graphene [12]. By introducing intricate
patterns and tailored structures, GK overcomes the low stretchability barrier
of graphene, resulting in unique mechanical properties [13, 14]. Compared to a
regular graphene sheet, GK nanostructures exhibit significantly improved
reversible stretchability, achieved through multidimensional deformation
capabilities enabled by the nanoscale Kirigami architectures [15]. The GK’s
in-plane stretching and out-of-plane deformation allow for increased
flexibility [15]. With flexible devices being the mainstream trend in modern
optoelectronics, Kirigami-like materials have become ideal candidates for
developing novel applications.

The fabrication of GK structures has yielded remarkable success [12, 16, 17,
18, 19, 20, 21, 22, 23, 24]. Initial work by Blees and colleagues demonstrated
the feasibility of GKs, showcasing robust microscale structures with
adjustable mechanical properties in graphene [12]. Another experimental study
highlighted the strain-insensitive electrical properties of GK-based
electrodes, enduring up to 240% tensile strain and various strain states,
including stretching, twisting, and shearing [21]. Furthermore, atomistic
molecular dynamics (MD) simulations have extensively investigated GK’s
stretchability, fracture resistance, and permeability [17, 13, 14, 15, 25, 26,
18, 19, 27]. Notably, these studies unveiled a noteworthy trend: GK can
enhance the fracture strains of graphene by approximately threefold compared
to standard monolayer graphene. Consequently, the realm of Kirigami-inspired
nanostructures presents exciting prospects for the fabrication of highly
flexible and stretchable devices using graphene [28] and other diverse 2D
materials [29, 30, 31].

A temperature change induced by external strain, known as the elastocaloric
effect (ECE), has great potential for next-generation thermal management
technologies due to its environmental friendliness and economic benefits [32,
33, 34]. To make this technology more efficient, efforts have been made to
find highly efficient elastocaloric materials [35, 36, 37, 38, 39, 40]. In
particular, in view of the growing interest in miniaturized refrigeration
[41], at the nanoscale, the study of the ECE in nanostructures has already
begun [42, 43, 44, 45, 46, 47, 48, 49, 50]. Recently, a new approach to
modulate the elastocaloric temperature change using a Kirigami-inspired
material was proposed [51]. Results showed that the ultrahigh stretchability
of kirigami combined with its lateral flexibility allows for the generation of
focused heat absorption and release simultaneously during the same application
of tensile or compressive strain, resulting in a versatile local
cooling/heating performance [51]. The method can be applied to any
elastocaloric material and pave the way for designing versatile solid-state
heat pumps for flexible electronics. So far, this phenomenon in Kirigami has
not been studied on the atomic scale. Moreover, based on the successful
synthesis of GKs, a study on the ECE in these materials may open new channels
for developing novel thermal management technologies at the nanoscale.

In this study, we employed fully-atomistic MD simulations to investigate the
ECE in a model GK lattice at the nanoscale. To simulate the ECE in GKs, we
modeled a thermal management device under Otto-like thermodynamic cycles,
including two adiabatic expansion/contraction and two isochoric heat exchange
processes. Using our computational protocol, we calculate the GK temperature
variation in absorbing and releasing heat and their coefficients of
performance.

##  2 Methodology

To investigate the ECE in a model GK lattice at the atomic scale, we used the
LAMMPS [52] code with the Adaptive Intermolecular Reactive Empirical Bond
Order (AIREBO) [53] potential in reactive MD simulations. This potential
accurately addresses carbon-based nanostructures and provides reliable
potential energy values to study their mechanical and thermal properties [54,
55, 56, 57, 58, 59]. Besides, the LAMMPS/AIREBO methodology has been
successfully used to describe the ECE in carbon nanomaterials [42, 45, 47, 50,
48]. The GK model used in this study is shown in Figure 1 and has the
following properties: $l_{0}=332$ Å, $l_{1}=48$ Å, $l_{2}=4.8$ Å, $h_{0}=100$
Å, $h_{1}=68$ Å, $h_{2}=35$ Å, $d_{f}=19$ Å, 12012 carbon atoms, and 1244
hydrogen atoms.

Figure 1: Schematic representation of the model GK lattice studied here.
Panels (a-c) show the GK top view, lattice parameters definitions, and KG side
view. In the color scheme, the grey and white spheres represent the carbon and
hydrogen atoms, respectively.

The GK structure was studied under a thermodynamic cycle similar to the Otto
cycle, as shown in Figure 2. The cycle consists of two adiabatic
expansion/release processes and two volume fixed heat exchange processes with
a thermal reservoir at 300 K, in the order $1\rightarrow 2\rightarrow
3\rightarrow 4\rightarrow 1$. Processes $1\rightarrow 2$ and $3\rightarrow 4$
correspond to applying adiabatic tensile strain and release, respectively.
Processes $2\rightarrow 3$ and $4\rightarrow 1$ represent isochoric heat
exchange with a reservoir at 300 K. During the stretching/compression process,
the left edge of the kirigami remained fixed, while the right edge was
stretched/compressed. In the isochoric heat exchange, both edges remained
fixed.

Figure 2: Schematic representation of a P-V-like diagram that illustrates the
thermodynamic cycle ($1\rightarrow 2\rightarrow 3\rightarrow 4\rightarrow 1$)
used to study the ECE in GK. Processes $1\rightarrow 2$ and $3\rightarrow 4$
correspond to the application of adiabatic tensile strain and release,
respectively, while processes $2\rightarrow 3$ and $4\rightarrow 1$ represent
isochoric heat exchange with a reservoir at 300 K.

Initially, we minimized the GK structure using a force and relative energy
tolerance of $1.0\times 10^{-10}$ eV/Å and $1.0\times 10^{-10}$, respectively.
The initial GK length ($l_{0}$, refer to Figure 1) is 337.20 Å, and its
equilibration was then conducted at 300 K, using a constant NPT ensemble
integration at null pressure using the Nosé-Hoover barostat for 200 ps to
ensure no remaining stress. The thermodynamic cycle was performed as follows:
first, we carried out step ($1\rightarrow 2$, expansion) without a thermostat,
subjecting the system to tensile strain from 0% to 30% at a strain rate of
$1\cdot 10^{-4}$ fs-1 during 32 ps, using a time step of 0.02 fs. Second, step
($2\rightarrow 3$, isochoric heat exchange) involved a 200 ps thermal
equilibration using the Langevin thermostat and NVE ensemble for a time step
of 0.2 fs. Step ($3\rightarrow 4$, release) was performed without a
thermostat, causing the system to undergo tensile strain from 30% to 0% at a
strain rate of $-1\cdot 10^{-4}$ fs-1 during 32 ps, using a time step of 0.02
fs again. Finally, step ($4\rightarrow 1$, isochoric heat exchange) involved
another 200 ps thermal equilibration using the Langevin thermostat and NVE
ensemble for a time step of 0.2 fs.

The real ECE temperature variation ($\Delta T_{\text{REAL}}$) is obtained from
correcting the results from classical MD simulations due to quantum mechanics
effects on the specific heat ($c$) of graphene at 300 K [42]: $\Delta
T_{\text{REAL}}=(c_{\text{MD}}/c_{\text{REAL}})\Delta T_{\text{MD}}$. Here,
“REAL” and “MD” refer to corrected and simulated values, respectively. The
$(c_{\text{MD}}/c_{\text{REAL}})=2.53$ factor was obtained using
$c_{\text{REAL}}=700$ $\mathrm{J}\cdot\mathrm{Kg}^{-1}\cdot\mathrm{K}^{-1}$
from Ref. [60] and $c_{\text{MD}}=1770$
$\mathrm{J}\cdot\mathrm{Kg}^{-1}\cdot\mathrm{K}^{-1}$ from our calculations.
$c_{\text{MD}}$ was derived from a series of equilibrium simulations for GK
subjected to distinct temperature regimes, as discussed later.

##  3 Results

We begin our discussion by showing the representative MD snapshots
illustrating GK’s stretching/releasing process (see Figure 3). In Figure 3(a),
we present the top and side views for the KG structure at 0% of strain and
equilibrated at 300 K. Even considering the atomic displacements imposed by
the temperature effects, one can note that the GK is almost planar and its
pores have the same size. It is worth mentioning that between 0%-5% of strain,
the GK deformation occurs via C-C elastic bond stretching in the direction of
the applied strain, similar to the stretching mechanism of graphene
nanoribbons [61]. In this low-strain regime, neither out-of-plane flipping nor
rotation of the GK pieces is observed.

Figure 3: Representative MD snapshots for the GK lattice subjected to the
uniaxial strain in the $x$-direction at $300$ K. Panels (a-e) illustrate the
top and side views for the GK structure at 0% of strain before stretching, 15%
of stretching, 30% of stretching, 15% of releasing, and 0% of strain after
releasing, respectively. In the color scheme, the grey and white spheres
represent the carbon and hydrogen atoms, respectively.

At a strain of 15% (see Figure 3(b)), noticeable flipping and rotation
occurred in certain sections of GK, resulting in varying magnitudes of out-of-
plane deflections instead of the anticipated continuous bond stretching in
uniformly strained sheets [62]. Unlike pristine graphene nanoribbons that
experience failure when reaching a critical strain [63], these deflections
enable the GK monolayer to withstand significantly larger deformations without
fracturing.

It is crucial to contrast our findings with the study conducted by Hirai and
coworkers [51], who investigated the electromechanical cooling effect (EMCE)
in various macroscopic kirigami materials (KMs). Using sophisticated lock-in
thermography techniques, they measured temperature changes across the kirigami
surface and discovered that cooling and warming regions coexist within
different regions of the structure during the same type of straining (tensile
or compressive). This phenomenon arises from the bending of hole edges within
the kirigami plane under tensile strain, inducing both tensile and compressive
strains in the outer and inner regions of the bent structure. They also
demonstrated that the cooled and warmed regions precisely corresponded to the
tensile-strained and compressed parts of the kirigami, respectively. However,
we should point out that this effect does not (and cannot) occur in one-atom-
thick kirigami.

In our simulations, depicted in Figures 3(b-d), the deformation mechanism of
GK involves two distinct regions: elastic stretching of C-C bonds (resulting
in an ECE increase of temperature) and out-of-plane buckling. Unlike
macroscopic kirigami, these buckled regions neither experienced cooling nor
warming. This behavior diverges from the EMCE observed in macroscopic KMs. The
reason behind this disparity is relatively straightforward. When a thin but
macroscopic ribbon flexes, it induces tensile and compressive strains in its
outer and inner sections. However, flexing a one-atom-thick nanoribbon, as
demonstrated in previous research [64], does not lead to any bond strain.
Therefore, in our case, the out-of-plane deflections did not generate strains
in the C-C bonds.

The accumulation of stress in GK is supported by the von Mises stress
distribution calculated from an MD study of GK under tension [15].
Consequently, as neither tensile nor compressive strains occur in the out-of-
plane deflected regions of GK, there is no heat release or absorption in these
areas. In contrast, the heated regions consist of parallel lines of stretched
C-C bonds that remain within the GK plane [15]. This stretching mechanism
observed in GK opposes the findings of experiments [51], where stretching
elastocaloric materials with a macroscopic Kirigami structure resulted in
simultaneous heat absorption and release due to non-uniform internal stress
generated by uneven strain distribution along the cutting patterns.

Figure 3(c) shows that the highly stretchable GK pattern, which exhibits no
bond breaking or fracture, is achieved at a critical strain of 30%. In this
configuration, each GK unit is connected by a small ribbon, allowing the
monolayer to be almost foldable. Additionally, a uniform out-of-plane
deflection pattern is observed. This configuration limits the interplay
between the warming and cooling of GK via ECE temperature variation. Figures
3(d) and 3(e) illustrate the GK monolayer configuration during the tensile-
releasing mechanism, which features are similar to those described for Figures
3(a) and 3(b), respectively.

We conducted ten simulations with different seeds for all the thermodynamic
cycles, see Supplementary Material. For the sake of convenience and to present
the qualitative behavior of the entire process, we show the time evolution of
the GK temperature under tensile stretching and releasing for just one seed in
Figures 4(a) and 4(b), respectively. This figure’s red and blue lines
represent the moving averages for temperature calculated every 100 MD steps.
We obtained the moving averages for all seeds and related average values for
$\Delta T_{\text{MD}}$ and used the $(c_{\text{MD}}/c_{\text{REAL}})=2.53$
factor to find $\Delta T_{\text{REAL}}\simeq 9.32$ K during the $1\rightarrow
2$ tensile stretching process, and $\Delta T_{\text{REAL}}\simeq-3.50$ K for
the $3\rightarrow 4$ releasing process. The smoothed curves indicate that the
GK temperature has an almost linear relationship with time. Additionally, it
can be inferred from Figure 4 that the ECE in GK is more significant during
heating than during cooling.

Figure 4: Time evolution of the GK temperature under (a) tensile stretching
and (b) releasing for one particular seed. The red and blue lines are the
moving averages for temperature calculated every 100 MD steps. Panel (c)
illustrates the time evolution of total energy as a function of time. The red
and blue curves refer to adiabatic tensile strain and the tensile strain
release, respectively.

The GK coefficient-of-performance (COP) associated with heating or cooling can
be calculated. To do that, we need to determine both the heat, $Q$, exchanged
with a thermal bath and the work, $W$, done on and by the system during one
full cycle. Processes $2\rightarrow 3$ and $4\rightarrow 1$ are isochoric and
do not contribute to the total work per cycle, while processes $1\rightarrow
2$ and $3\rightarrow 4$ are adiabatic, meaning there is no heat exchange into
or out of the GK monolayer. Therefore, the total energy change due to tensile
or tensile-release processes will equal the work done on or by the GK system.
The total energy values as a function of simulation time during the
$1\rightarrow 2$ tensile stretching process and the $3\rightarrow 4$ releasing
process are shown in Figure 4(c), also just for one seed. $W$ can, then, be
calculated from these curves as the sum of the variations in the internal
energy of the GK during processes $1\rightarrow 2$ and $3\rightarrow 4$,
denoted as $E_{1\rightarrow 2}$ and $E_{3\rightarrow 4}$, respectively.
Specifically, their average values considering all the seeds are
$W=E_{1\rightarrow 2}-E_{3\rightarrow 4}=109.5-(-103.1)=212.6$ eV.

To obtain the heat, $Q$, exchanged with the thermal bath, the thermal
capacity, $C$, of the GK was estimated using a series of equilibrium
simulations at different temperature regimes, as shown in Figure 5. The
thermalization of the GK over time and the relationship between the
temperature of the system and its total energy are shown in Figures 5(a) and
5(b), respectively. Temperature values ranged from 50 K to 1000 K in
increments of 150 K. Linear fitting of the data in Figure 5(b) resulted in a
thermal capacity of $C_{\text{MD}}=\Delta E/\Delta T=2.67$ eV/K, which divided
by the mass of the GK and changing the units give the value previously
mentioned of $c_{\text{MD}}=1770$
$\mathrm{J}\cdot\mathrm{Kg}^{-1}\cdot\mathrm{K}^{-1}$. With $c_{\text{MD}}$
and the values of $\Delta T_{\text{MD}}$, one can calculate the heat
transferred/extracted ($Q_{\text{T}}$/$Q_{\text{E}}$) to/from the thermal
reservoir at 300 K: $Q_{\text{E}}=mc_{\text{MD}}\Delta T_{\text{MD}}$ and
$Q_{\text{T}}=mc_{\text{MD}}\Delta T_{\text{MD}}$.

Figure 5: (a) Time evolution of the temperature (GK thermalization) and (b)
the total energy of the system as a function of its temperature. The
temperature values used in these simulations vary from 50 K up to 1100 K with
a step of 150 K.

Finally, we calculate the COP for GK. Using the values derived above for $Q$
and $W$, we obtain the COP for heat transfer and extraction
(COP${}_{\text{T}}$/COP${}_{\text{E}}$) to/from the thermal reservoir:
COP${}_{\text{T}}=Q_{\text{T}}/W_{\text{MD}}$ and
COP${}_{\text{E}}=Q_{\text{E}}/W_{\text{MD}}$. Some alloys, for instance, NiTi
wires, show an ECE of about 17 K with a large COP of about 11 [65]. Carbon
nanotubes can exhibit an ECE varying between 40 K to 100 K with also a large
COP that ranges from 4 to 12 [48]. GK has reasonable temperature change values
of approximately 9.32 K for heating and -3.50 K for cooling, as mentioned
above, accompanied by an equally reasonable COP of approximately 1.57 and
0.62, respectively. Although these values are not so impressive as that of
NiTi wires or carbon nanotubes, they represent about 23 and 8.8 times larger
ECE temperature changes than that of macroscopic kirigami materials [51]. The
Supplementary Material provides the values for the aforementioned quantities
for individual seeds, which were used to determine the COP.

##  4 Conclusion

In summary, our study investigated the ECE in a GK monolayer at the nanoscale,
representing the thinnest possible KM. We compared the temperature changes of
GK with macroscopic KMs, which was experimentally realized recently [51], and
identified key differences. Moreover, this study is the first to determine the
COP in GK.

Through fully-atomistic molecular dynamics simulations, we found that GK
exhibited a relatively large temperature change of approximately 9.32 K for
heating and -3.50 K for cooling. This temperature change is significantly
higher than the observed temperature changes in macroscopic KMs, which reached
a maximum of approximately 0.4 K. The one-atom-thick configuration of graphene
in GK prevented the complex temperature variations observed in macroscopic
KMs, where simultaneous heating and cooling occurred in different regions when
the external strain was applied.

Our simulations also revealed that the deformation mechanism of GK involved
elastic stretching of C-C bonds and out-of-plane buckling. The out-of-plane
deflected regions did not experience cooling or warming, contrary to the
behavior observed in macroscopic KMs. This disparity arose because flexing a
one-atom-thick nanoribbon did not generate strains in the C-C bonds, unlike
macroscopic kirigami structures.

We calculated the COP of GK, representing the ratio of heat transferred
to/from the thermal reservoir to work performed on the system per
thermodynamic cycle. Our results demonstrated a reasonable COP of
approximately 1.57 (0.62) for GK as a solid heater (refrigerator), indicating
its potential for heat transfer and extraction. When comparing the COP of GK
to that of other materials, such as alloys and carbon nanotubes, GK presents a
smaller performance. However, neither alloys nor nanotubes reversibly
withstand a tensile strain as high as 30%. That feature might be considered
when considering smart applications of GK as solid heaters or refrigerators at
the nanoscale. Our findings highlight the unique behavior of GK at the
nanoscale and its potential as a kirigami-based material for applications
requiring substantial temperature changes and efficient heat transfer.

## Acknowledgements

This work received partial support from Brazilian agencies CAPES, CNPq, and
FAPDF. L.A.R.J thanks the financial support from Brazilian Research Council
FAP-DF grants $00193-00000857/2021-14$, $00193-00000853/2021-28$, and
$00193-00000811/2021-97$, CNPq grants $302236/2018-0$ and $350176/2022-1$, and
FAPDF-PRONEM grant $00193.00001247/2021-20$. L.A.R.J also thanks ABIN grant
08/2019 and Núcleo de Computação de Alto Desempenho (NACAD) for computational
facilities through the Lobo Carneiro supercomputer. AFF is a fellow of the
Brazilian Agency CNPq-Brazil ($303284/2021-8$) and acknowledges grant
#$2020/02044-9$ from São Paulo Research Foundation (FAPESP). This work used
resources of the Centro Nacional de Processamento de Alto Desempenho em São
Paulo (CENAPAD-SP). The authors acknowledge the National Laboratory for
Scientific Computing (LNCC/MCTI, Brazil) for providing HPC resources of the
SDumont supercomputer, contributing to the research results reported within
this paper. URL: http://sdumont.lncc.br.

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