Commit dfb16030 authored by Zdenek Dvorak's avatar Zdenek Dvorak
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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 $j<i$ 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}
......
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