Commit dfb16030 by Zdenek Dvorak

### 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 $i=0,\l ... @@ -306,23 +306,92 @@ Moreover,$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 $f(u)\cap f(v)\neq\emptyset$. $xy\in E(T_\beta)$. Hence, again we have $f(u)\cap f(v)\neq\emptyset$. \end{proof} \end{proof} Note that in Theorem~\ref{thm-cs}, we only get a representation of a supergraph We can now combine Theorem~\ref{thm-cs} with Lemma~\ref{lemma-cliq}. of $G$. \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 %We will need the fact that the chromatic number is at most exponential in the comparable box dimension; $G\subseteq G'$ and $\cbdim(G')\le (\cbdim(\GG)+1)\bigl(2^{\cbdim(\GG)}+1\bigr)\le 6^{\cbdim(\GG)}$. %this follows from~\cite{subconvex} and we include the argument to make the dependence clear. \end{corollary} %\begin{lemma}\label{lemma-chrom} %If $G$ has a comparable box representation $f$ in $\mathbb{R}^d$, then $G$ is $3^d$-colorable. Note that only bound the comparable box dimension of a supergraph %\end{lemma} of $G$. To deal with this issue, we show that the comparable box dimension of a subgraph %\begin{proof} is at most exponential in the comparable box dimension of the whole graph. %We actually show that $G$ is $(3^d-1)$-degenerate. Since every induced subgraph of $G$ also This is essentially Corollary~25 in~\cite{subconvex}, but since the setting is somewhat %has a comparable box representation in $\mathbb{R}^d$, it suffices to show that the minimum degree of $G$ different and the construction of~\cite{subconvex} uses rotated boxes, %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 $v$, we provide details of the argument. %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 \emph{star coloring} of a graph $G$ is a proper coloring such that any two color classes induce %a box obtained from $f(v)$ by scaling it by a factor of three, and thus $1+|N(v)|\le 3^d$. a star forest (i.e., a graph not containing any 4-vertex path). The \emph{star chromatic number} $\chi_s(G)$ %\end{proof} 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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