Commit 0d75f91f by Zdenek Dvorak

### Some minor changes to the introduction.

parent 1cc4c115
 ... ... @@ -49,8 +49,8 @@ Torsten Ueckerdt\thanks{Karlsruhe Institute of Technology. E-mail: {\tt torsten \maketitle \begin{abstract} Two boxes in $\mathbb{R}^d$ are comparable if one of them is a subset of a translation of the other. The comparable box dimension of a graph Two boxes in $\mathbb{R}^d$ are \emph{comparable} if one of them is a subset of a translation of the other one. The \emph{comparable box dimension} of a graph $G$ is the minimum integer $d$ such that $G$ can be represented as a touching graph of comparable axis-aligned boxes in $\mathbb{R}^d$. We show that proper minor-closed classes have bounded comparable box ... ... @@ -68,16 +68,17 @@ This result has motivated numerous strengthenings and variations (see \cite{grap 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}). This is in contrast to general intersection representations where the represented class always includes arbitrarily large cliques. Of course, whether the class of touching graph of objects from $\OO$ is sparse or not depends on the system $\OO$. For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of axis-aligned boxes in $\mathbb{R}^3$, where the vertices in Of course, whether the class of touching graphs of objects from $\OO$ is sparse or not depends on the system $\OO$. For example, all complete bipartite graphs $K_{n,m}$ are touching graphs of boxes in $\mathbb{R}^3$, where the vertices in 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). boxes (throughout the paper, by a \emph{box} we mean an axis-aligned one, i.e., the Cartesian product of closed intervals of non-zero length). 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, 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. 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 comparable box dimension of graphs in $\GG$ is not bounded. ... ... @@ -86,7 +87,7 @@ they showed that if a class $\GG$ has finite comparable box dimension, then it h 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} proved that graphs of comparable box dimension $3$ have exponential weak coloring number, giving the proved that graphs of comparable box dimension $3$ have exponential weak coloring numbers, giving the first natural graph class with polynomial strong coloring numbers and superpolynomial weak coloring numbers (the previous example is obtained by subdividing edges of every graph suitably many times~\cite{covcol}). ... ... @@ -103,10 +104,10 @@ This gives arbitrarily precise approximation algorithms for all monotone maximiz expressible in terms of distances between the solution vertices and tractable on graphs of bounded treewidth~\cite{distapx} or expressible in the first-order logic~\cite{logapx}. \section{Parameters} 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 $\cbdim(G)$. In this section we bound some basic graph parameters in terms of comparable box dimension. The first result bounds the clique number $\omega(G)$ in terms of $\cbdim(G)$. \begin{lemma}\label{lemma-cliq} For any graph $G$, we have $\omega(G)\le 2^{\cbdim(G)}$. \end{lemma} ... ... @@ -121,6 +122,9 @@ least one of the $2^d$ orthants at $p$. At the same time, it follows from the de of a touching representation that $f(a_1),\ldots,f(a_d)$ have pairwise disjoint interiors, and hence $w \leq 2^d$. \end{proof} Note that a clique with $2^d$ vertices has a touching representation by comparable boxes in $\mathbb{R}^d$, 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 ... ... @@ -129,7 +133,7 @@ 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} although we include an 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 $G$ we have $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot ... ... @@ -139,14 +143,14 @@ For any graph$G$we have$\chi(G)\le 3^{\cbdim(G)}$and$\chi_s(G) \le 2\cdot 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 $f(v_i)$ $v_j$ such that $ji$ 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 $f(v_i)$ 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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