@@ -79,7 +79,7 @@ A \emph{touching representation by comparable boxes} of a graph $G$ is a touchin
such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable.
Let the \emph{comparable box dimension}$\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.
Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details.
Then for a class $\GG$ of graphs, let $\cbdim(\GG):=\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the
Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the
comparable box dimension of graphs in $\GG$ is not bounded.
Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular,
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@@ -126,21 +126,25 @@ Note that a clique with $2^d$ vertices has a touching representation by comparab
where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$.
Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$.
In the following we consider the chromatic number $\chi(G)$, and one
of its variants. A \emph{star coloring} of a graph $G$ is a proper
coloring such that any two color classes induce a star forest (i.e., a
graph not containing any 4-vertex path). The \emph{star chromatic
number}$\chi_s(G)$ of $G$ is the minimum number of colors in a star
coloring of $G$. We will need the fact that the star chromatic number
is at most exponential in the comparable box dimension; this follows
In the following we consider the chromatic number $\chi(G)$, and two
of its variants. An \emph{acyclic coloring} (resp.\emph{star coloring}) of a graph $G$ is a proper
coloring such that any two color classes induce a forest (resp. star forest, i.e., a
graph not containing any 4-vertex path). The \emph{acyclic chromatic number}$\chi_a(G)$ (resp. \emph{star chromatic
number}$\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star)
coloring of $G$. We will need the fact that all the variants of the chromatic number
are at most exponential in the comparable box dimension; this follows
from~\cite{subconvex}, although we include an argument to make the
dependence clear.
\begin{lemma}\label{lemma-chrom}
For any graph $G$ we have $\chi(G)\le3^{\cbdim(G)}$ and $\chi_s(G)\le2\cdot
9^{\cbdim(G)}$.
For any graph $G$ we have $\chi(G)\le3^{\cbdim(G)}$, $\chi_a(G)\le5^{\cbdim(G)}$ and $\chi_s(G)\le2\cdot9^{\cbdim(G)}$.
\end{lemma}
\begin{proof}
We focus on the star chromatic number and note that the chromatic number may be bounded similarly. Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$ be the vertices of $G$ written so that $\vol(f(v_1))\geq\ldots\geq\vol(f(v_n))$. Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy colouring $c$ so that $c(i)$ is the smallest color such that $c(i)\neq c(j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path.
We focus on the star chromatic number and note that the chromatic number and the acyclic chromatic number may be bounded similarly.
Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$
be the vertices of $G$ written so that $\vol(f(v_1))\geq\ldots\geq\vol(f(v_n))$.
Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy colouring $c$ so that $c(i)$ is
the smallest color such that $c(i)\neq c(j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there
exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path.
It remains to bound the number of colors used. Suppose we are coloring $v_i$. We shall bound the number of vertices
$v_j$ such that $j<i$ and there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
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@@ -516,14 +520,12 @@ clique-sums.
\end{itemize}
\end{proof}
The following lemma shows that any graphs has a $C^\star$-clique-sum
extendable representation in $\mathbb{R}^d$, for $d=\omega(G)+
\ecbdim(G)$ and for any clique $C^\star$.
The following lemma enables us to pick the root clique at the expense of increasing
the dimension by $\omega(G)$.
\begin{lemma}\label{lem-apex-cs}
For any graph $G$ and any clique $C^\star$, we have that $G$ admits a
$C^\star$-clique-sum extendable touching representation by comparabe
boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| +\ecbdim(G\setminus
V(C^\star))$.
For any graph $G$ and any clique $C^\star$, the graph $G$ admits a
$C^\star$-clique-sum extendable touching representation by comparable
boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| +\ecbdim(G\setminus V(C^\star))$.
\end{lemma}
\begin{proof}
The proof is essentially the same as the one of
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@@ -532,40 +534,43 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
comparable boxes in $\mathbb{R}^{d'}$, with $d' =\cbdim(G\setminus
V(C^\star))$, and let $V(C^\star)=\{v_1,\ldots,v_k\}$. We now construct
the desired representation $h$ of $G$ as follows. For each vertex
$v_i\in V(C^\star)$ let $h(v_i)$ be the box fulfilling (v1) with
$d_{v_i}= i$. For each vertex $u\in V(G)\setminus V(C^\star)$, if $i\le
k$ then let $h(u)[i]=[0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i]=
$v_i\in V(C^\star)$, let $h(v_i)$ be the box in $\mathbb{R}^d$ uniquely determined
by the condition (v1) with $d_{v_i}= i$. For each vertex $u\in V(G)\setminus V(C^\star)$,
if $i\lek$ then let $h(u)[i]=[0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i]=
[1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i]=
\alpha_i h'(u)[i-k]$, for some $\alpha_i>0$. The values$\alpha_i>0$
are chosen suffciently small so that $h(u)[i]\subset[0,1)$, whenever $u\notin V(C^\star)$.
\alpha h'(u)[i-k]$, for some $\alpha>0$. The value $\alpha>0$
is chosen suffciently small so that $h(u)[i]\subset[0,1)$ whenever $u\notin V(C^\star)$.
We proceed similarly for the clique points. For any
clique $C$ of $G$, if $i\le k$ then let $p(C)[i]=0$ if $v_i \in
V(C)$, and $p(C)[i]=1/4$ if $v_i \notin V(C)$. For $i>k$ we have
to refer to the clique point $p'(C')$ of $C'=C\setminus
\{v_1,\ldots,v_k\}$, as we set $p(C)[i]=\alpha_i p'(C')[i-k]$.
clique $C$ of $G$, if $i\le k$ then let $p(C)[i]=0$ if $v_i \in V(C)$,
and $p(C)[i]=1/4$ if $v_i \notin V(C)$. For $i>k$ we refer to the clique point $p'(C')$ of $C'=C\setminus
\{v_1,\ldots,v_k\}$, and we set $p(C)[i]=\alpha p'(C')[i-k]$.
By the construction, it is clear that $h$ is a touching representation of $G$.
As $h'(u)\sqsubset h'(v)$ implies that $h(u)\sqsubset h(v)$, and as
$h(u)\sqsubset h(v_i)$, for every $u\in V(G)\setminus V(C^\star)$ and every
$v_i \in V(C^\star)$, we have that $h$ is a touching representation by comparable boxes.
By the construction, it is clear that $h$ is a representation of $G$.
For the $C^\star$-clique-sum extendability, it is clear that the (vertices) conditions hold.
For the (cliques) condition (c1), let us first consider two distinct cliques $C_1$ and $C_2$
$h(u)\sqsubset h(v_i)$ for every $u\in V(G)\setminus V(C^\star)$ and every
$v_i \in V(C^\star)$, we have that $h$ is a representation by comparable boxes.
For the $C^\star$-clique-sum extendability, the \textbf{(vertices)} conditions hold by the construction.
For the \textbf{(cliques)} condition (c1), let us consider distinct cliques $C_1$ and $C_2$
of $G$ such that $|V(C_1)| \ge |V(C_2)|$, and let $C'_i=C_i\setminus V(C^\star)$. If $C'_1= C'_2$,
there is a vertex $v_i \in V(C_1)\setminus V(C_2)$, and $p(C_1)[i]=0\neq1/4= p(C_2)[i]$.
Otherwise, if $C'_1\neq C'_2$, we have that$p'(C'_1)\neq p'(C'_2)$, which leads to
Otherwise, if $C'_1\neq C'_2$, then$p'(C'_1)\neq p'(C'_2)$, which implies
$p(C_1)\neq p(C_2)$ by construction.
For the (cliques) condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ containing $v$.
In the first dimensions $i \le k$, we always have $h^\varepsilon(C)[i]\subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$ we have
$h^\varepsilon(C)[i]\subseteq[0,1/2]= h(v)[i]$ (as in that case $v$ and $v_i$ are adjacent), and if $v_i \notin V(C)$
we have $h^\varepsilon(C)[i]\subseteq[1/4,1/2]\subseteq h(v)[i]$. Then for the last $d'$ dimensions, by definition of $h'$,
we have that $h^\varepsilon(C)[i]\subseteq h(v)[i]$ for every $i>k$, except one,
for which $h^\varepsilon(C)[i]\cap h(v)[i]=\{p(C)[i]\}$. This completes the first case
and we now consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
As $v\notin V(C')$, there is an hyperplane
${\mathcal H}' =\{ p\in\mathbb{R}^{d'}\ |\ p[i]= c\}$ that separates $p'(C')$ and $h'(v)$.
This implies that the following hyperplane
${\mathcal H}=\{ p\in\mathbb{R}^{d}\ |\ p[k+i]=\alpha_{k+i} c\}$ separates $p(C)$ and $h(v)$.
Now we consider a vertex $v_i \in V(C^\star)$, and we note that for any clique $C$ containing $v_i$
For the \textbf{(cliques)} condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and
a clique $C$ of $G$ containing $v$. In the dimensions $i\in\{1,\ldots,k\}$, we always have
$h^\varepsilon(C)[i]\subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$, then
$h^\varepsilon(C)[i]\subseteq[0,1/2]= h(v)[i]$, as in this case $v$ and $v_i$ are adjacent;
and if $v_i \notin V(C)$, then $h^\varepsilon(C)[i]\subseteq[1/4,1/2]\subseteq h(v)[i]$.
By the property (c2) of $h'$,
we have $h^\varepsilon(C)[i]\subseteq h(v)[i]$ for every $i>k$, except one,
for which $h^\varepsilon(C)[i]\cap h(v)[i]=\{p(C)[i]\}$.
Next, let us consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
As $v\notin V(C')$, the condition (c2) for $h'$ implies that $p'(C')$ is disjoint from $h'(v)$,
and thus $p(C)$ is disjoint from $h(v)$.
Finally, we consider a vertex $v_i \in V(C^\star)$. Note that for any clique $C$ containing $v_i$,
we have that $h^\varepsilon(C)[i]\cap h(v_i)[i]=[0,\varepsilon]\cap[-1,0]=\{0\}$, and $h^\varepsilon(C)[j]\subseteq[0,1]= h(v_i)[j]$
for any $j\neq i$. For a clique $C$ that does not contain $v_i$ we have that
