From 24d20534cae220daa3845aff6cf3033ea621e6ce Mon Sep 17 00:00:00 2001
From: "jane.tan" <jane.tan@maths.ox.ac.uk>
Date: Thu, 10 Feb 2022 00:03:03 +0000
Subject: [PATCH] edits to address subset of referee report comments
---
comparable-box-dimension.tex | 17 ++++++++---------
1 file changed, 8 insertions(+), 9 deletions(-)
diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index 4ed3241..19591f1 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -98,7 +98,7 @@ Famously, Koebe~\cite{koebe} proved that a graph is planar if and only if it is
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}).
+(e.g., planar graphs, or more generally, graph classes with bounded expansion~\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 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
@@ -160,8 +160,7 @@ 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 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
+coloring such that any two color classes induce a forest (resp. star forest, i.e., a forest in which each component is a star). 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
@@ -206,7 +205,7 @@ is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedil
\node [fill=none, color=blue] at (4, 4) {$B_1$};
\node [fill=white] at (-0.8, 3) {$f(v_2)$};
\node [fill=none, color=blue] at (0.7, 3.3) {$B_2$};
- \node [fill=white] at (3.4, 1) {$f(v_3)$};
+ \node [fill=white] at (3.1, 1) {$f(v_3)$};
\node [fill=none, color=blue] at (2, 1) {$B_3$};
\end{tikzpicture}
\caption{Nearby boxes obstructing colors at $v_i$.}
@@ -352,7 +351,7 @@ If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+2\cdot 81^{\
\end{corollary}
Let us remark that an exponential increase in the dimension is unavoidable: We have $\cbdim(K_{2^d})=d$,
-but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. Indeed, for every pair $u,v$ of non-adjacent vertices there is a specific dimension $i$ such that their boxes span intervals $[a,b]$ and $[c,d]$ with $c<d$, while for every other box in the representation their $i^\text{th}$ interval contains $[b,c]$.
+but the graph obtained from $K_{2^d}$ by deleting a perfect matching has comparable box dimension $2^{d-1}$. Indeed, for every pair $u,v$ of non-adjacent vertices there is a specific dimension $i$ such that their boxes span intervals $[a,b]$ and $[c,d]$ with $b<c$, while for every other box in the representation their $i^\text{th}$ interval contains $[b,c]$.
\subsection{Clique-sums}
@@ -606,10 +605,10 @@ the point $p(C)$ for every clique $C$ as follows:
By construction, it is clear that for each vertex $v\in V(H)$, $p(C) \in h(v)$ if and only if
$v\in V(C)$.
-For any two distinct cliques $C_1$ and $C_2$ the points $p(C_1)$ and
-$p(C_2)$ are distinct. Indeed, by symmetry we can assume that for some $i$,
+For any two distinct cliques $C_1$ and $C_2$, the points $p(C_1)$ and
+$p(C_2)$ are distinct. Indeed, by symmetry we can assume that for some $i$
we have $v_i\in V(C_1)\setminus V(C_2)$, and this implies that $p(C_1)[i] < p(C_2)[i]$.
-Hence, the condition (v1) holds.
+Hence, the condition (c1) holds.
Consider now a vertex $v_i$ and a clique $C$. As we observed before, if $v_i\not\in V(C)$,
then $p(C) \not\in h(v_i)$, and thus $h^\varepsilon(C)$ and $h(v_i)$ are disjoint (for sufficiently small $\varepsilon>0$).
@@ -782,7 +781,7 @@ is an $a$-cell. Then $C$ is an open box with sides of lengths $\ell_{a,1}$, \ld
\begin{itemize}
\item If $a=1$, then $\ell_{a,j}=ksd |\omega(v_a)[j]|$.
\item If $a>1$ and $\ell_{a,j}=\ell_{a-1,j}$, then $\ell_{a,j}=\ell_{a-1,j}<2ksd|\omega(v_a)[j]|$ (otherwise $\ell_{a,j}=\ell_{a-1,j}/b$ for some integer $b\ge 2$).
-\item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integers $b\ge 2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that
+\item If $a>1$ and $\ell_{a,j} < \ell_{a-1,j}$, then $\ell_{a-1,j}\ge b\times ksd|\omega(v_a)[j]|$ for some integer $b\ge 2$. Now let $b$ be the greatest such integer (that is such that $\ell_{a-1,j} < (b+1)\times ksd|\omega(v_a)[j]|$) and note that
\[\ell_{a,j}=\frac{\ell_{a-1,j}}{b}<\tfrac{b+1}{b}ksd|\omega(v_a)[j]|<\tfrac{3}{2}ksd|\omega(v_a)[j]|.\]
\end{itemize}
Hence, $\ell_{a,j}<2ksd |\omega(v_a)[j]|$. Let $C'$ be the box with the same center as $C$ and with $|C'[j]|=(2ksd+2)|\omega(v_a)[j]|$.
--
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