Some minor changes to the introduction.

comparable-box-dimension.tex

@@ -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 $j<i$ 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 $f(v_i)$
+$v_j$ 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 $f(v_i)$
 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 $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedily, we only need to avoid colors of at most
 $$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$
-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 $B_j$}
+vertices. 
 \end{proof}
 
 \begin{figure}
@@ -184,7 +188,7 @@ subgraph $H$ of $G$ by simply deleting the boxes corresponding to the
 vertices in $V(G)\setminus V(H)$.  In this section we are going to
 consider other basic operations on graphs.
 
-\subsection{Vertex addition}
+\subsection{Vertex addition}\label{sec-vertad}
 
 Let us start with a simple lemma saying that the addition of a vertex
 increases the comparable box dimension by at most one.  In particular,