Commit 1cc4c115 by Jane Tan

### Changes for clarity in Section Parameters

parent 9c39d3a6
 ... ... @@ -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}). ... ... @@ -74,16 +73,16 @@ For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis one part are represented by $m\times 1\times 1$ boxes and the vertices of the other part are represented by $1\times n\times 1$ 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} ... ... @@ -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 $\omega(G)\le 2^{\cbdim(G)}$. For any graph $G$, we have $\omega(G)\le 2^{\cbdim(G)}$. \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 $w \leq 2^d$. 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 $w \leq 2^d$. \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 $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot For any graph$G$we have$\chi(G)\le 3^{\cbdim(G)}$and$\chi_s(G) \le 2\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$such that$jj$such that$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_1i$such that$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)$for any$jj$such that$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_1i$for which$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
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