### The subgraph argument.

parent 0951065d
 ... ... @@ -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$jj$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$\vol(B_j)/\vol(f(v_i))-1=5^d-1$. A similar argument shows that the number of indices$m$such that$m
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