The subgraph argument.

comparable-box-dimension.tex

@@ -306,23 +306,92 @@ Moreover, $x\prec y$, implying that $E_i(u)\cap E_i(v)\neq\emptyset$ for $i=0,\l
 $xy\in E(T_\beta)$.  Hence, again we have $f(u)\cap f(v)\neq\emptyset$.
 \end{proof}
 
-Note that in Theorem~\ref{thm-cs}, we only get a representation of a supergraph
-of $G$.  
-
-%We will need the fact that the chromatic number is at most exponential in the comparable box dimension;
-%this follows from~\cite{subconvex} and we include the argument to make the dependence clear.
-%\begin{lemma}\label{lemma-chrom}
-%If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$-colorable.
-%\end{lemma}
-%\begin{proof}
-%We actually show that $G$ is $(3^d-1)$-degenerate.  Since every induced subgraph of $G$ also
-%has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$
-%is less than $3^d$.  Let $v$ be a vertex of $G$ such that $f(v)$ has the smallest volume.  For every neighbor $u$ of $v$,
-%there exists a translation $B_u$ of $f(v)$ such that $B_u\subseteq f(u)$ and $B_u$ touches $f(v)$.
-%Note that $f(v)\cup \bigcup_{u\in N(v)} B_u$ is a union of internally disjoint translations of $f(v)$ contained in
-%a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+|N(v)|\le 3^d$.
-%\end{proof}
+We can now combine Theorem~\ref{thm-cs} with Lemma~\ref{lemma-cliq}.
+\begin{corollary}\label{cor-cs}
+If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then there exists a graph $G'$ such that
+$G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}$.
+\end{corollary}
+
+Note that only bound the comparable box dimension of a supergraph
+of $G$.  To deal with this issue, we show that the comparable box dimension of a subgraph
+is at most exponential in the comparable box dimension of the whole graph.
+This is essentially Corollary~25 in~\cite{subconvex}, but since the setting is somewhat
+different and the construction of~\cite{subconvex} uses rotated boxes,
+we provide details of the argument.
+
+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 dependence clear.
+\begin{lemma}\label{lemma-chrom}
+If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ has star chromatic number at most $2\cdot 9^d$.
+\end{lemma}
+\begin{proof}
+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 $\vol(B_j)/\vol(f(v_i))-1=5^d-1$.
+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
+$$(5^d-1) + (3^d-1) + (3^d-1)^2<5^d+9^d<2\cdot 9^d$$
+vertices.
+\end{proof}
+
+Next, let us show a bound on the comparable box dimension of subgraphs.
+
+\begin{lemma}\label{lemma-subg}
+If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+\chi^2_s(G')$.
+\end{lemma}
+\begin{proof}
+As we can remove the boxes that represent the vertices, we can assume $V(G')=V(G)$.
+Let $f$ be a touching representation by comparable boxes in $\mathbb{R}^d$, where $d=\cbdim(G')$.  Let $\varphi$
+be a star coloring of $G'$ using colors $\{1,\ldots,c\}$, where $c=\chi_s(G')$.
+
+For any distinct colors $i,j\in\{1,\ldots,c\}$, let $A_{i,j}\subseteq V(G)$ consist of vertices $u$ of color $i$
+such that there exists a vertex $v$ of color $j$ such that $uv\in E(G')$ and $uv\not\in E(G)$.
+Let us define a representation $h$ by boxes in $\mathbb{R}^{d+\binom{c}{2}}$ by starting from the representation $f$ and,
+for each pair $i<j$ of colors, adding a dimension $d_{i,j}$ and setting
+$h(v)[d_{i,j}]=[1/3,4/3]$ for $v\in A_{i,j}$, $h(v)[d_{i,j}]=[-4/3,-1/3]$ for $v\in A_{j,i}$,
+and $h(v)[d_{i,j}](v)=[-1/2,1/2]$ otherwise.  Note that the boxes in this extended representation are comparable,
+as in the added dimensions, all the boxes have size $1$.
+
+Suppose $uv\in E(G)$, where $\varphi(u)=i$ and $\varphi(v)=j$ and say $i<j$.  The boxes $f(u)$ and $f(v)$ touch.
+We cannot have $u\in A_{i,j}$ and $v\in A_{j,u}$, as then $G'$ would contain a 4-vertex path in colors $i$ and $j$.
+Hence, in any added dimension $d'$, at least one of $h(u)$ and $h(v)$ is represented by the interval $[-1/2,1/2]$,
+and thus $h(u)[d']\cap h(v)[d']\neq\emptyset$.  Therefore, the boxes $h(u)$ and $h(v)$ touch.
+
+Suppose now that $uv\not\in E(G)$.  If $uv\not\in E(G')$, then $f(u)$ is disjoint from $f(v)$, and thus $h(u)$ is disjoint from
+$h(v)$.  Hence, we can assume $uv\in E(G')$, $\varphi(u)=i$, $\varphi(v)=j$ and $i<j$.  Then $u\in A_{i,j}$, $v\in A_{j,i}$,
+$h(u)[d_{i,j}]=[1/3,4/3]$, $h(v)[d_{j,i}]=[-4/3,-1/3]$, and $h(u)\cap h(v)=\emptyset$.
+
+Consequently, $h$ is a touching representation of $G$ by comparable boxes in dimension $d+\binom{c}{2}\le d+c^2$.
+\end{proof}
+
+Let us now combine Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}.
+
+\begin{corollary}\label{cor-subg}
+If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le \cbdim(G')+4\cdot 81^{\cbdim(G')}\le 5\cdot 81^{\cbdim(G')}$.
+\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 $2^{d-1}$.
+Corollaries~\ref{cor-cs} and~\ref{cor-subg} now give the main result of this section.
 
+\begin{corollary}\label{cor-comb}
+If $G$ is obtained from graphs in a class $\GG$ by clique-sums, then 
+$\cbdim(G)\le 5\cdot 81^{6^{\cbdim(\GG)}}$.
+\end{corollary}
 
 \section{The product structure and minor-closed classes}