@@ -59,12 +59,11 @@ dimension and explore further properties of this notion.

\section{Introduction}

For a system $\OO$ of subsets of $\mathbb{R}^d$, we say that a graph $G$ is a \emph{touching graph of objects from $\OO$}

Given a system $\OO$ of subsets of $\mathbb{R}^d$, we say that a graph $G$ is a \emph{touching graph of objects from $\OO$}

if there exists a function $f:V(G)\to\OO$ (called a \emph{touching representation by objects from $\OO$})

such that for distinct $u,v\in V(G)$, the interiors of $f(u)$ and $f(v)$ are disjoint

and $f(u)\cap f(v)\neq\emptyset$ if and only if $uv\in E(G)$.

such that the interiors of $f(u)$ and $f(v)$ are disjoint for all distinct $u,v\in V(G)$, and $f(u)\cap f(v)\neq\emptyset$ if and only if $uv\in E(G)$.

Famously, Koebe~\cite{koebe} proved that a graph is planar if and only if it is a touching graph of balls in $\mathbb{R}^2$.

This result motivated a number of strengthenings and variations (see \cite{graphsandgeom, sachs94} for some classical examples); most relevantly for us, every planar graph is a touching graph of cubes in $\mathbb{R}^3$~\cite{felsner2011contact}.

This result has motivated numerous strengthenings and variations (see \cite{graphsandgeom, sachs94} for some classical examples); most relevantly for us, Felsner and Francis~\cite{felsner2011contact} showed that every planar graph is a touching graph of cubes in $\mathbb{R}^3$.

An attractive feature of touching representations is that it is possible to represent graph classes that are sparse

(e.g., planar graphs, or more generally, graph classes with bounded expansion theory~\cite{nesbook}).

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@@ -74,16 +73,16 @@ For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis

one part are represented by $m\times1\times1$ boxes and the vertices of the other part are represented by $1\times n\times1$

boxes (a \emph{box} is the Cartesian product of intervals of non-zero length, in particular axis-aligned).

Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} noticed that this issue disappears if we forbid such a combination of

long and wide boxes: For two boxes $B_1$ and $B_2$, we write $B_1\sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.

long and wide boxes, a condition which can be expressed as follows. For two boxes $B_1$ and $B_2$, we write $B_1\sqsubseteq B_2$ if $B_2$ contains a translate of $B_1$.

We say that $B_1$ and $B_2$ are \emph{comparable} if $B_1\sqsubseteq B_2$ or $B_2\sqsubseteq B_1$.

A \emph{touching representation by comparable boxes} of a graph $G$ is a touching representation $f$ by boxes

such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. For a graph $G$, let the \emph{comparable box dimension}$\cbdim(G)$

of$G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.

For a class $\GG$ of graphs, let $\cbdim(\GG)=\sup\{\cbdim(G):G\in\GG\}$; note that $\cbdim(\GG)=\infty$ if the

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 $\mathbb{R}^d$.

Then for a class $\GG$ of graphs, let $\cbdim(\GG):=\sup\{\cbdim(G):G\in\GG\}$. 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,

they proved that if a class $\GG$ has finite comparable box dimension, then it has polynomial strong coloring

they showed that if a class $\GG$ has finite comparable box dimension, then it has polynomial strong coloring

numbers, which implies that $\GG$ has strongly sublinear separators. They also provided an example showing

that for any function $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite

comparable box dimension. Dvo\v{r}\'ak et al.~\cite{wcolig}

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@@ -107,57 +106,75 @@ or expressible in the first-order logic~\cite{logapx}.

\section{Parameters}

Let us first bound the clique number $\omega(G)$ in terms of

$\cbdim(G)$.

In this section we bound some basic graph parameters in terms of comparable box dimension. Since the statements are trivial for graphs of unbounded comparable box dimension, we need not consider them in the proofs. The first result bounds the clique number $\omega(G)$ in terms of $\cbdim(G)$.

\begin{lemma}\label{lemma-cliq}

For any graph $G$, then$\omega(G)\le2^{\cbdim(G)}$.

For any graph $G$, we have$\omega(G)\le2^{\cbdim(G)}$.

\end{lemma}

\begin{proof}

To represent any clique $A =\{a_1,\ldots,a_w\}$ in $G$, the

We may assume that $G$ has bounded comparable box dimension

witnessed by a representation $f$. To represent any clique $A =\{a_1,\ldots,a_w\}$ in $G$, the

corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty

intersection. Since axis-aligned boxes have the Helly property, there

is a point $p \in\mathbb{R}^d$ contained in $f(a_1)\cap\cdots\cap

f(a_w)$. As each box is full-dimensional, its interior intersects at

least one of the $2^d$ orthants at $p$. Since $f$ is a touching

representation,$f(a_1),\ldots,f(a_d)$ have pairwise disjoint

interiors and hence $w \leq2^d$.

least one of the $2^d$ orthants at $p$. At the same time, it follows from the definition

of a touching representation that$f(a_1),\ldots,f(a_d)$ have pairwise disjoint

interiors, and hence $w \leq2^d$.

\end{proof}

In the following we consider the chromatic number $\chi(G)$, and one

of its variant. A \emph{star coloring} of a graph $G$ is a proper

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

from~\cite{subconvex} and we include the argument to make the

from~\cite{subconvex} although we include an argument to make the

dependence clear.

\begin{lemma}\label{lemma-chrom}

For any graph $G$, then$\chi(G)\le3^{\cbdim(G)}$ and $\chi_s(G)\le2\cdot

For any graph $G$ we have$\chi(G)\le3^{\cbdim(G)}$ and $\chi_s(G)\le2\cdot

9^{\cbdim(G)}$.

\end{lemma}

\begin{proof}

Let us focus on the star chromatic number.

Let $v_1$, \ldots, $v_n$ be the vertices of $G$ ordered non-increasingly by the size of the boxes that represent them;

i.e., so that $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. We greedily color the vertices in order, giving $v_i$ the smallest

color different from the colors of all vertices $v_j$ such that $j<i$ and either $v_jv_i\in E(G)$, or there exists $m>j$

such that $v_jv_m,v_mv_i\in E(G)$. Note this gives a star coloring: A path on four vertices always contains a 3-vertex subpath

$v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$, and in such a path, the coloring procedure gives each vertex a distinct color.

Hence, it remains to bound the number of colors we used. Let us fix some $i$, and let us first bound the number of vertices

$v_j$ such that $j<i$ and there exists $m>i$ such that $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box that is five times larger than $f(v)$

and has the same center as $f(v)$. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$

contained in $f(v_j)\cap B$. The boxes $B_j$ for different $j$ have disjoint interiors and their interiors are also disjoint from

$f(v_i)\subset B$, and thus the number of such indices $j$ is at most $\vol(B_j)/\vol(f(v_i))-1=5^d-1$.

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.

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 centre. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$

contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B_j)/\vol(f(v_i))-1=5^d-1$.

A similar argument shows that the number of indices $m$ such that $m<i$ and $v_mv_i\in E(G)$ is at most $3^d-1$.

Consequently, the number of indices $j<i$ for which there exists $m$ such that $j<m<i$ and $v_jv_m,v_mv_i\in E(G)$

is at most $(3^d-1)^2$.

Consequently, when choosing the color of $v_i$ greedily, we only need to avoid colors of at most

is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedily, we only need to avoid colors of at most

$$(5^d-1)+(3^d-1)+(3^d-1)^2<5^d+9^d<2\cdot9^d$$

vertices. We proceed similarly to bound the chromatic number.

vertices. \note{J: Why isn't $5^d-1$ enough by itself? We only worry about vertices in the 2-ball around $v_i$, and it seems that for each such $v_j$ with boxes bigger than $f(v_i)$ we can find a translate $B_j$}

\end{proof}

\begin{figure}

\centering

\begin{tikzpicture}[xscale=1, yscale=0.6]

\footnotesize

\draw[dashed] (0,0) rectangle (5,5);

\filldraw[red!20!white] (2,2) rectangle (3,3);

\draw (1,0) rectangle (3,2);

\node [fill=none, color=red] at (2.5, 2.5) {$f(v_i)$};