@@ -117,8 +117,8 @@ comparable box dimension of graphs in $\GG$ is not bounded.
Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular,
they showed that if a class $\GG$ has finite comparable box dimension, then it has polynomial strong coloring
numbers, which implies that $\GG$ has strongly sublinear separators. They also provided an example showing
that for any function $h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
comparable box dimension. Dvo\v{r}\'ak et al.~\cite{wcolig}
that for many functions$h$, the class of graphs with strong coloring numbers bounded by $h$ has infinite
comparable box dimension\footnote{In their construction $h(r)$ has to be at least 3, and has to tend to $+\infty$.}. Dvo\v{r}\'ak et al.~\cite{wcolig}
proved that graphs of comparable box dimension $3$ have exponential weak coloring numbers, giving the
first natural graph class with polynomial strong coloring numbers and superpolynomial weak coloring numbers
(the previous example is obtained by subdividing edges of every graph suitably many times~\cite{covcol}).
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@@ -174,19 +174,17 @@ For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G)
We focus on the star chromatic number and note that the chromatic number and the acyclic chromatic number may be bounded similarly.
Suppose that $G$ has comparable box dimension $d$ witnessed by a representation $f$, and let $v_1, \ldots, v_n$
be the vertices of $G$ written so that $\vol(f(v_1))\geq\ldots\geq\vol(f(v_n))$.
Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy coloring $c$ so that $c(i)$ is
the smallest color such that $c(i)\neq c(j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there
Equivalently, we have $f(v_i)\sqsubseteq f(v_j)$ whenever $i>j$. Now define a greedy coloring $c$ so that $c(v_i)$ is
the smallest color such that $c(v_i)\neq c(v_j)$ for any $j<i$ for which either $v_jv_i\in E(G)$ or there
exists $m>j$ such that $v_jv_m,v_mv_i\in E(G)$. Note that this gives a star coloring, since a path on four vertices always contains a 3-vertex subpath of the form $v_{i_1}v_{i_2}v_{i_3}$ such that $i_1<i_2,i_3$ and our coloring procedure gives distinct colors to vertices forming such a path.
It remains to bound the number of colors used. Suppose we are coloring $v_i$. We shall bound the number of vertices
$v_j$ such that $j<i$ and there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B_j)/\vol(f(v_i))-1=5^d-1$.
$v_j$ such that $j<i$ and such that there exists $m>i$ for which $v_jv_m,v_mv_i\in E(G)$. Let $B$ be the box obtained by scaling up $f(v_i)$ by a factor of 5 while keeping the same center. Since $f(v_m)\sqsubseteq f(v_i)\sqsubseteq f(v_j)$, there exists a translation $B_j$ of $f(v_i)$
contained in $f(v_j)\cap B$ (see Figure~\ref{fig:lowercolcount}). Two boxes $B_{j}$ and $B_{j'}$ for $j\neq j'$ have disjoint interiors since their intersection is contained in the intersection of the touching boxes $f(v_{j})$ and $f(v_{j'})$, and their interiors are also disjoint from $f(v_i)\subset B$. Thus the number of such indices $j$ is at most $\vol(B)/\vol(f(v_i))-1=5^d-1$.
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 $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedily, we only need to avoid colors of at most
\[(5^d-1)+(3^d-1)+(3^d-1)^2<5^d+9^d<2\cdot9^d\]
vertices.
is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedily, we only need to avoid colors of at most $(5^d-1)+(3^d-1)+(3^d-1)^2$ vertices, so $2\cdot9^d$ colors suffice.
\end{proof}
\begin{figure}
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@@ -222,7 +220,8 @@ It is clear that given a touching representation of a graph $G$, one
easily obtains a touching representation by boxes of an induced
subgraph $H$ of $G$ by simply deleting the boxes corresponding to the
vertices in $V(G)\setminus V(H)$. In this section we are going to
consider other basic operations on graphs.
consider other basic operations on graphs. In the following, to describe
the boxes, we are going to use the Cartesian product $\times$ defined among boxes ($A\times B$ is the box whose projection on the first dimensions gives the box $A$, while the projection on the remaing dimensions gives the box $B$) or we are going to provide its projections for every dimension ($A[i]$ is the interval obtained from projecting $A$ on its $i^\text{th}$ dimension).
\subsection{Vertex addition}\label{sec-vertad}
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@@ -312,7 +311,7 @@ different and the construction of~\cite{subconvex} uses rotated boxes,
we provide details of the argument.
\begin{lemma}\label{lemma-subg}
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le\cbdim(G')+\chi^2_s(G')$.
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le\cbdim(G')+\frac12\chi^2_s(G')$.
\end{lemma}
\begin{proof}
As we can remove the boxes that represent the vertices, we can assume $V(G')=V(G)$.
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@@ -343,17 +342,17 @@ Suppose now that $uv\not\in E(G)$. If $uv\not\in E(G')$, then $f(u)$ is disjoin
$h(v)$. Hence, we can assume $uv\in E(G')\setminus E(G)$, $\varphi(u)=i$, $\varphi(v)=j$ 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 $d+\binom{c}{2}\le d+c^2$.
Consequently, $h$ is a touching representation of $G$ by comparable boxes in dimension $d+\binom{c}{2}\le d+c^2/2$.
\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 $\cbdim(G)\le\cbdim(G')+4\cdot81^{\cbdim(G')}\le5\cdot81^{\cbdim(G')}$.
If $G$ is a subgraph of a graph $G'$, then $\cbdim(G)\le\cbdim(G')+2\cdot81^{\cbdim(G')}\le3\cdot81^{\cbdim(G')}$.
\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 $2^{d-1}$.
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 $[b,c]$.
\subsection{Clique-sums}
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@@ -381,20 +380,19 @@ by (not necessarily comparable) boxes in $\mathbb{R}^d$ is called
\begin{itemize}
\item[]{\bf(vertices)} For each $u\in V(C^\star)$, there exists a dimension $d_u$,
such that:
\subitem[(v0)]$d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
\subitem[(v1)] each vertex $u\in V(C^\star)$ satisfies $h(u)[d_u]=[-1,0]$ and
\subitem\emph{(v0)}$d_u\neq d_{u'}$ for distinct $u,u'\in V(C^\star)$,
\subitem\emph{(v1)} each vertex $u\in V(C^\star)$ satisfies $h(u)[d_u]=[-1,0]$ and
$h(u)[i]=[0,1]$ for any dimension $i\neq d_u$, and
\subitem[(v2)] each vertex $v\notin V(C^\star)$ satisfies $h(v)\subset[0,1)^d$.
\subitem\emph{(v2)} each vertex $v\notin V(C^\star)$ satisfies $h(v)\subset[0,1)^d$.
\item[]{\bf(cliques)} For every clique $C$ of $G$, there exists
a point \[p(C)\in[0,1)^d\cap\bigcap_{v\in V(C)} h(v)\]
a point $p(C)\in[0,1)^d\cap\left(\bigcap_{v\in V(C)} h(v)\right)$
such that, defining the \emph{clique box}$h^\varepsilon(C)$
by setting
\[h^\varepsilon(C)[i]=[p(C)[i],p(C)[i]+\varepsilon]\] for every dimension
by setting $h^\varepsilon(C)[i]=[p(C)[i],p(C)[i]+\varepsilon]$ for every dimension
$i$, the following conditions are satisfied:
\begin{itemize}
\subitem[(c1)] For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1)\cap
% \begin{itemize}
\subitem\emph{(c1)} For any two cliques $C_1\neq C_2$, $h^\varepsilon(C_1)\cap
Similarly, we can deal with proper minor-closed classes.
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@@ -816,7 +643,7 @@ Let $\GG$ be a proper minor-closed class. Since $\GG$ is proper, there exists $
By Theorem~\ref{thm-prod}, there exists $k$ such that every graph in $\GG$ is a subgraph of a graph obtained by repeated clique-sums
from extended $k$-tree-grids. As we have seen, $k$-tree-grids have comparable box dimension at most $k+2$,
and by Lemma~\ref{lemma-apex}, extended $k$-tree-grids have comparable box dimension at most $2k+2$.
By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le625^{2k+2}$.
By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le1250^{2k+2}$.
\end{proof}
Note that the graph obtained from $K_{2n}$ by deleting a perfect matching has Euler genus $\Theta(n^2)$
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@@ -897,7 +724,7 @@ is fractionally treewidth-fragile, with a function $f(k) = O_{t,s,d}\bigl(k^{d}\
For a positive integer $k$, let $f(k)=(2ksd+2)^dst$.
Let $(\iota,\omega)$ be an $s$-comparable envelope representation of a graph $G$
in $\mathbb{R}^d$ of thickness at most $t$, and let $v_1$, \ldots, $v_n$ be the corresponding ordering of the vertices of $G$.
Let us define $\ell_{i,j}\in\mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $|ksd\omega(v_i)[j]|$ such that $\ell_{i-1,j}/\ell_{i,j}$ is a positive integer. Formally
Let us define $\ell_{i,j}\in\mathbb{R}^+$ for $i=1,\ldots, n$ and $j\in\{1,\ldots,d\}$ as an approximation of $ksd|\omega(v_i)[j]|$ such that $\ell_{i-1,j}/\ell_{i,j}$ is a positive integer. Formally
it is defined as follows.
\begin{itemize}
\item Let $\ell_{1,j}=ksd|\omega(v_1)[j]|$.
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@@ -989,4 +816,206 @@ has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr).
\end{corollary}
\bibliography{data}
\appendix
\section{Omitted proofs}
\newtheorem*{lemma-A}{Lemma~\ref{lem-cs}}
\begin{lemma-A}
Consider two graphs $G_1$ and $G_2$, given with a $C^\star_1$- and a
$C^\star_2$-clique-sum extendable representations $h_1$ and $h_2$ by comparable boxes
in $\mathbb{R}^{d_1}$ and $\mathbb{R}^{d_2}$,
respectively. Let $G$ be the graph obtained by performing a full
clique-sum of these two graphs on any clique $C_1$ of $G_1$, and on
the root clique $C^\star_2$ of $G_2$. Then $G$ admits a $C^\star_1$-clique
sum extendable representation $h$ by comparable boxes in
$\mathbb{R}^{\max(d_1,d_2)}$.
\end{lemma-A}
\begin{proof}
By Lemma~\ref{lemma-add}, we can assume that $d_1=d_2$; let $d=d_1$.
The idea is to translate (allowing also exchanges of dimensions) and
scale $h_2$ to fit in $h_1^\varepsilon(C_1)$. Consider an $\varepsilon >0$
sufficiently small so that $h_1^\varepsilon(C_1)$ satisfies all the
\textbf{(cliques)} conditions, and such that $h_1^\varepsilon(C_1)\sqsubseteq
h_1(v)$ for any vertex $v\in V(G_1)$. Let $V(C_1)=\{v_1,\ldots,v_k\}$;
without loss of generality, we can assume $i_{C_1,v_i}=i$ for $i\in\{1,\ldots,k\}$,
