Commit dcce432f authored by Zdenek Dvorak's avatar Zdenek Dvorak
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Revision of the rest of the paper.

parent 5ada1937
...@@ -79,7 +79,7 @@ A \emph{touching representation by comparable boxes} of a graph $G$ is a touchin ...@@ -79,7 +79,7 @@ A \emph{touching representation by comparable boxes} of a graph $G$ is a touchin
such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable. such that for every $u,v\in V(G)$, the boxes $f(u)$ and $f(v)$ are comparable.
Let the \emph{comparable box dimension} $\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$. Let the \emph{comparable box dimension} $\cbdim(G)$ of a graph $G$ be the smallest integer $d$ such that $G$ has a touching representation by comparable boxes in $\mathbb{R}^d$.
Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details. Let us remark that the comparable box dimension of every graph $G$ is at most $|V(G)|$, see Section~\ref{sec-vertad} for details.
Then for a class $\GG$ of graphs, let $\cbdim(\GG):=\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the Then for a class $\GG$ of graphs, let $\cbdim(\GG)\colonequals\sup\{\cbdim(G):G\in\GG\}$. Note that $\cbdim(\GG)=\infty$ if the
comparable box dimension of graphs in $\GG$ is not bounded. 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, Dvo\v{r}\'ak, McCarty and Norin~\cite{subconvex} proved some basic properties of this notion. In particular,
...@@ -126,21 +126,25 @@ Note that a clique with $2^d$ vertices has a touching representation by comparab ...@@ -126,21 +126,25 @@ Note that a clique with $2^d$ vertices has a touching representation by comparab
where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$. where each vertex is a hypercube defined as the Cartesian product of intervals of form $[-1,0]$ or $[0,1]$.
Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$. Together with Lemma~\ref{lemma-cliq}, it follows that $\cbdim(K_{2^d})=d$.
In the following we consider the chromatic number $\chi(G)$, and one In the following we consider the chromatic number $\chi(G)$, and two
of its variants. A \emph{star coloring} of a graph $G$ is a proper of its variants. An \emph{acyclic coloring} (resp. \emph{star coloring}) of a graph $G$ is a proper
coloring such that any two color classes induce a star forest (i.e., a coloring such that any two color classes induce a forest (resp. star forest, i.e., a
graph not containing any 4-vertex path). The \emph{star chromatic graph not containing any 4-vertex path). The \emph{acyclic chromatic number} $\chi_a(G)$ (resp. \emph{star chromatic
number} $\chi_s(G)$ of $G$ is the minimum number of colors in a star number} $\chi_s(G)$) of $G$ is the minimum number of colors in an acyclic (resp. star)
coloring of $G$. We will need the fact that the star chromatic number coloring of $G$. We will need the fact that all the variants of the chromatic number
is at most exponential in the comparable box dimension; this follows are at most exponential in the comparable box dimension; this follows
from~\cite{subconvex}, although we include an argument to make the from~\cite{subconvex}, although we include an argument to make the
dependence clear. dependence clear.
\begin{lemma}\label{lemma-chrom} \begin{lemma}\label{lemma-chrom}
For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot For any graph $G$ we have $\chi(G)\le 3^{\cbdim(G)}$, $\chi_a(G)\le 5^{\cbdim(G)}$ and $\chi_s(G) \le 2\cdot 9^{\cbdim(G)}$.
9^{\cbdim(G)}$.
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
We focus on the star chromatic number and note that the 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 colouring $c$ 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 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. 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 colouring $c$ 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
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 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 $f(v_i)$ $v_j$ 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 $f(v_i)$
...@@ -516,14 +520,12 @@ clique-sums. ...@@ -516,14 +520,12 @@ clique-sums.
\end{itemize} \end{itemize}
\end{proof} \end{proof}
The following lemma shows that any graphs has a $C^\star$-clique-sum The following lemma enables us to pick the root clique at the expense of increasing
extendable representation in $\mathbb{R}^d$, for $d= \omega(G) + the dimension by $\omega(G)$.
\ecbdim(G)$ and for any clique $C^\star$.
\begin{lemma}\label{lem-apex-cs} \begin{lemma}\label{lem-apex-cs}
For any graph $G$ and any clique $C^\star$, we have that $G$ admits a For any graph $G$ and any clique $C^\star$, the graph $G$ admits a
$C^\star$-clique-sum extendable touching representation by comparabe $C^\star$-clique-sum extendable touching representation by comparable
boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$.
V(C^\star))$.
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
The proof is essentially the same as the one of The proof is essentially the same as the one of
...@@ -532,40 +534,43 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) + ...@@ -532,40 +534,43 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
comparable boxes in $\mathbb{R}^{d'}$, with $d' = \cbdim(G\setminus comparable boxes in $\mathbb{R}^{d'}$, with $d' = \cbdim(G\setminus
V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We now construct V(C^\star))$, and let $V(C^\star) = \{v_1,\ldots,v_k\}$. We now construct
the desired representation $h$ of $G$ as follows. For each vertex the desired representation $h$ of $G$ as follows. For each vertex
$v_i\in V(C^\star)$ let $h(v_i)$ be the box fulfilling (v1) with $v_i\in V(C^\star)$, let $h(v_i)$ be the box in $\mathbb{R}^d$ uniquely determined
$d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$, if $i\le by the condition (v1) with $d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^\star)$,
k$ then let $h(u)[i] = [0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i] = if $i\le k$ then let $h(u)[i] = [0,1/2]$ if $uv_i \in E(G)$, and $h(u)[i] =
[1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i] = [1/4,3/4]$ if $uv_i \notin E(G)$. For $i>k$ we have $h(u)[i] =
\alpha_i h'(u)[i-k]$, for some $\alpha_i>0$. The values $\alpha_i>0$ \alpha h'(u)[i-k]$, for some $\alpha>0$. The value $\alpha>0$
are chosen suffciently small so that $h(u)[i] \subset [0,1)$, whenever $u\notin V(C^\star)$. is chosen suffciently small so that $h(u)[i] \subset [0,1)$ whenever $u\notin V(C^\star)$.
We proceed similarly for the clique points. For any We proceed similarly for the clique points. For any
clique $C$ of $G$, if $i\le k$ then let $p(C)[i] = 0$ if $v_i \in clique $C$ of $G$, if $i\le k$ then let $p(C)[i] = 0$ if $v_i \in V(C)$,
V(C)$, and $p(C)[i] = 1/4$ if $v_i \notin V(C)$. For $i>k$ we have and $p(C)[i] = 1/4$ if $v_i \notin V(C)$. For $i>k$ we refer to the clique point $p'(C')$ of $C'=C\setminus
to refer to the clique point $p'(C')$ of $C'=C\setminus \{v_1,\ldots,v_k\}$, and we set $p(C)[i] = \alpha p'(C')[i-k]$.
\{v_1,\ldots,v_k\}$, as we set $p(C)[i] = \alpha_i p'(C')[i-k]$.
By the construction, it is clear that $h$ is a touching representation of $G$.
As $h'(u) \sqsubset h'(v)$ implies that $h(u) \sqsubset h(v)$, and as As $h'(u) \sqsubset h'(v)$ implies that $h(u) \sqsubset h(v)$, and as
$h(u) \sqsubset h(v_i)$, for every $u\in V(G)\setminus V(C^\star)$ and every $h(u) \sqsubset h(v_i)$ for every $u\in V(G)\setminus V(C^\star)$ and every
$v_i \in V(C^\star)$, we have that $h$ is a touching representation by comparable boxes. $v_i \in V(C^\star)$, we have that $h$ is a representation by comparable boxes.
By the construction, it is clear that $h$ is a representation of $G$.
For the $C^\star$-clique-sum extendability, it is clear that the (vertices) conditions hold. For the $C^\star$-clique-sum extendability, the \textbf{(vertices)} conditions hold by the construction.
For the (cliques) condition (c1), let us first consider two distinct cliques $C_1$ and $C_2$ For the \textbf{(cliques)} condition (c1), let us consider distinct cliques $C_1$ and $C_2$
of $G$ such that $|V(C_1)| \ge |V(C_2)|$, and let $C'_i=C_i\setminus V(C^\star)$. If $C'_1 = C'_2$, of $G$ such that $|V(C_1)| \ge |V(C_2)|$, and let $C'_i=C_i\setminus V(C^\star)$. If $C'_1 = C'_2$,
there is a vertex $v_i \in V(C_1) \setminus V(C_2)$, and $p(C_1)[i] = 0 \neq 1/4 = p(C_2)[i]$. there is a vertex $v_i \in V(C_1) \setminus V(C_2)$, and $p(C_1)[i] = 0 \neq 1/4 = p(C_2)[i]$.
Otherwise, if $C'_1 \neq C'_2$, we have that $p'(C'_1) \neq p'(C'_2)$, which leads to Otherwise, if $C'_1 \neq C'_2$, then $p'(C'_1) \neq p'(C'_2)$, which implies
$p(C_1) \neq p(C_2)$ by construction. $p(C_1) \neq p(C_2)$ by construction.
For the (cliques) condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ containing $v$.
In the first dimensions $i \le k$, we always have $h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$ we have For the \textbf{(cliques)} condition (c2), let us first consider a vertex $v\in V(G)\setminus V(C^\star)$ and
$h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$ (as in that case $v$ and $v_i$ are adjacent), and if $v_i \notin V(C)$ a clique $C$ of $G$ containing $v$. In the dimensions $i\in\{1,\ldots,k\}$, we always have
we have $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. Then for the last $d'$ dimensions, by definition of $h'$, $h^\varepsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$, then
we have that $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one, $h^\varepsilon(C)[i] \subseteq [0,1/2] = h(v)[i]$, as in this case $v$ and $v_i$ are adjacent;
for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. This completes the first case and if $v_i \notin V(C)$, then $h^\varepsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$.
and we now consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$. By the property (c2) of $h'$,
As $v\notin V(C')$, there is an hyperplane we have $h^\varepsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
${\mathcal H}' = \{ p\in \mathbb{R}^{d'}\ |\ p[i] = c\}$ that separates $p'(C')$ and $h'(v)$. for which $h^\varepsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$.
This implies that the following hyperplane
${\mathcal H} = \{ p\in \mathbb{R}^{d}\ |\ p[k+i] = \alpha_{k+i} c\}$ separates $p(C)$ and $h(v)$. Next, let us consider a vertex $v\in V(G)\setminus V(C^\star)$ and a clique $C$ of $G$ not containing $v$.
Now we consider a vertex $v_i \in V(C^\star)$, and we note that for any clique $C$ containing $v_i$ As $v\notin V(C')$, the condition (c2) for $h'$ implies that $p'(C')$ is disjoint from $h'(v)$,
and thus $p(C)$ is disjoint from $h(v)$.
Finally, we consider a vertex $v_i \in V(C^\star)$. Note that for any clique $C$ containing $v_i$,
we have that $h^\varepsilon(C)[i] \cap h(v_i)[i] = [0,\varepsilon]\cap [-1,0] = \{0\}$, and $h^\varepsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$ we have that $h^\varepsilon(C)[i] \cap h(v_i)[i] = [0,\varepsilon]\cap [-1,0] = \{0\}$, and $h^\varepsilon(C)[j] \subseteq [0,1] = h(v_i)[j]$
for any $j\neq i$. For a clique $C$ that does not contain $v_i$ we have that for any $j\neq i$. For a clique $C$ that does not contain $v_i$ we have that
$h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$. $h^\varepsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$.
...@@ -578,184 +583,213 @@ of $\cbdim(G)$ and $\chi(G)$. ...@@ -578,184 +583,213 @@ of $\cbdim(G)$ and $\chi(G)$.
For any graph $G$, $\ecbdim(G) \le \cbdim(G) + \chi(G)$. For any graph $G$, $\ecbdim(G) \le \cbdim(G) + \chi(G)$.
\end{lemma} \end{lemma}
\begin{proof} \begin{proof}
Let $h$ be a touching representation by comparable boxes of $G$ in Let $h$ be a touching representation of $G$ by comparable boxes in
$\mathbb{R}^d$, with $d=\cbdim(G)$, and let $c$ be a $\mathbb{R}^d$, with $d=\cbdim(G)$, and let $c$ be a
$\chi(G)$-coloring of $G$. We start with a slightly modified version $\chi(G)$-coloring of $G$. We start with a slightly modified version
of $h$. We first scale $h$ to fit in $(0,1)^d$, and for a of $h$. We first scale $h$ to fit in $(0,1)^d$, and for a
sufficiently small real $\alpha>0$ we increase each box in $h$, by sufficiently small real $\alpha>0$ we increase each box in $h$ by
$2\alpha$ in every dimension, that is we replace $h(v)[i] = [a,b]$ $2\alpha$ in every dimension, that is we replace $h(v)[i] = [a,b]$
by $[a-\varepsilon,b+\varepsilon]$ for each vertex $v$ and dimension by $[a-\alpha,b+\alpha]$ for each vertex $v$ and dimension
$i$. Furthermore $\alpha$ is chosen sufficiently small, so that no $i$. We choose $\alpha$ sufficiently small so that the boxes representing
new intersection was created. The obtained representation $h_1$ is non-adjacent vertices remain disjoint, and thus the resulting representation $h_1$ is
thus an intersection representation of the same graph $G$ such that, an intersection representation of the same graph $G$. Moreover, observe that
for every clique $C$ of $G$, the intersection $I_C= \cap_{v\in V(C) for every clique $C$ of $G$, the intersection $I_C=\bigcap_{v\in V(C)} h_1(v)$ is
h_1(v)}$ is $d$-dimensional. For any maximal clique $C$ of $G$, let a box with non-zero edge lengths. For any clique $C$ of $G$, let
$p_1(C)$ be a point in the interior of $I_C$. $p_1(C)$ be a point in the interior of $I_C$ different from the points
chosen for all other cliques.
Now we add $\chi(G)$ dimensions to make the representation touching Now we add $\chi(G)$ dimensions to make the representation touching
again, and to ensure some space for the clique boxes again, and to ensure some space for the clique boxes
$h^\varepsilon(C)$. Formally we define $h_2$ as follows. $h^\varepsilon(C)$. Formally we define $h_2$ as follows.
$$h_2(u)[i]=\begin{cases} $$h_2(u)[i]=\begin{cases}
h_1(u)[i]&\text{ if $i\le d$}\\ h_1(u)[i]&\text{ if $i\le d$}\\
[1/5,3/5]&\text{ if $c(u) < i-d$}\\ [1/5,3/5]&\text{ if $i>d$ and $c(u) < i-d$}\\
[0,2/5]&\text{ if $c(u) = i-d$}\\ [0,2/5]&\text{ if $i>d$ and $c(u) = i-d$}\\
[2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)} [2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)}
\end{cases}$$ \end{cases}$$
For any clique $C'$ of $G$, let us denote $c(C')$, the color set For any clique $C$ of $G$, let $c(C)$ denote the color set $\{c(u)\ |\ u\in V(C)\}$.
$\{c(u)\ |\ u\in V(C')\}$, and let $C$ be one of the maximal cliques We now set
containing $C'$. We now set $$p_2(C)[i]=\begin{cases}
$$p_2(C')[i]=\begin{cases}
p_1(C) &\text{ if $i\le d$}\\ p_1(C) &\text{ if $i\le d$}\\
2/5 &\text{ if $i-d \in c(C')$}\\%\text{if $\exists u\in V(C)$ with $c(u) = i-d$}\\ 2/5 &\text{ if $i>d$ and $i-d \in c(C)$}\\
1/2 &\text{ otherwise} 1/2 &\text{ otherwise}
\end{cases} \end{cases}
$$ $$
As $h_2$ is an extension of $h_1$, and as for all the extra dimensions $j$ ($j>d$) As $h_2$ is an extension of $h_1$, and as in each dimension $j>d$,
we have that $2/5 \in h_2(v)[j]$ for every vertex $v$, we have that $h_2$ is an $h_2(v)[j]$ is an interval of length $2/5$ containing the point $2/5$ for every vertex $v$,
intersection representation of $G$. To prove that it is touching consider two adjacent we have that $h_2$ is an intersection representation of $G$ by comparable boxes.
To prove that it is touching consider two adjacent
vertices $u$ and $v$ such that $c(u)<c(v)$, and let us note that $h_2(u)[d+c(u)] = [0,2/5]$ vertices $u$ and $v$ such that $c(u)<c(v)$, and let us note that $h_2(u)[d+c(u)] = [0,2/5]$
and $h_2(v)[d+c(u)] = [2/5,4/5]$. By construction, the boxes of $h_1$ are comparable boxes, and $h_2(v)[d+c(u)] = [2/5,4/5]$.
and as $h_2$ is an extension of $h_1$ such that for all the extra dimensions $j$ ($j>d$)
the length of $h_2(v)[j]$ is $2/5$ for every vertex $v$, we have that the boxes in $h_2$ are For the $\emptyset$-clique-sum extendability, the \textbf{(vertices)} conditions are void.
comparables boxes. For the \textbf{(cliques)} conditions, since $p_1$ is chosen to be injective, the mapping $p_2$
For the $\emptyset$-clique-sum extendability, the (vertices) conditions clearly hold. is injective as well, implying that (c1) holds.
For the (cliques) conditions, let us first note that the points $p_1(C)$, defined for
the maximum cliques, are necessarily distinct. This impies that two cliques $C_1$ and $C_2$, Consider now a clique $C$ in $G$ and a vertex $v\in V(G)$. If $c(v)\not\in c(C)$, then
which clique points $p_2(C_1)$ and $p_2(C_2)$ are based on distinct maximum cliques, necessarily lead to distinct points. $h_2(v)[c(v)+d]=[0,2/5]$ and $p_2(C)[c(v)+d]=1/2$, implying that $h_2^{\varepsilon}(C)\cap h_2(v)=\emptyset$.
In the case that $C_1$ and $C_2$ belong to some maximal clique $C$, we have that $c(C_1) \neq c(C_2)$ If $c(v)\in c(C)$ but $v\not\in V(C)$, then letting $v'\in V(C)$ be the vertex of color $c(v)$,
and this implies by construction that $p_2(C_1)$ and $p_2(C_2)$ are distinct. Thus (c1) holds. we have $vv'\not\in E(G)$, and thus $h_1(v)$ is disjoint from $h_1(v')$. Since $p_1(C)$ is contained
By construction of $h_1$, we have that if $h_2^{\varepsilon}(C')[i] \cap h_2(v)[i]$ is non-empty for every $i\le d$, in the interior of $h_1(v')$, it follows that $h_2^{\varepsilon}(C)\cap h_2(v)=\emptyset$.
then we have that $h_2^{\varepsilon}(C')[i] \subset h_2(v)[i]$ for every $i\le d$, Finally, suppose that $v\in C$. Since $p_1(C)$ is contained in the interior of $h_1(v)$,
and we have that $v$ belongs to some maximal clique $C$ containing $C'$. If $v\notin V(C')$ note that we have $h_2^{\varepsilon}(C)[i] \subset h_2(v)[i]$ for every $i\le d$. For $i>d$ distinct from $d+c(v)$,
$p_2(C')[d+c(v)] = 1/2 \notin[0,2/5]=h_2(v)[d+c(v)]$, while if $v\in V(C')$ we have that we have $p_2^{\varepsilon}(C)[i]\in\{2/5,1/2\}$ and $[2/5,3/5]\subseteq h_2(v)[i]$, and thus
$h_2^{\varepsilon}(C')[i] \subset [2/5,1/2+\varepsilon] \subset h_2(v)[i]$ for every dimension $i>d$, $h_2^{\varepsilon}(C)[i] \subset h_2(v)[i]$. For $i=d+c(v)$, we have $p_2^{\varepsilon}(C)[i]=2/5$
except if $c(v)=i-d$, and in that case $h_2(v)[i] \cap h_2^{\varepsilon}(C')[i] = and $h_2(v)[i]=[0,2/5]$, and thus $h_2^{\varepsilon}(C)[i] \cap h_2(v)[i]=\{p_2^{\varepsilon}(C)[i]\}$.
[0,2/5]\cap[2/5,2/5+\varepsilon] = \{2/5\}$. We thus have that (c2) holds, and this concludes the proof of the lemma. Therefore, (c2) holds.
\end{proof} \end{proof}
Together, the lemmas from this section show that comparable box dimension is almost preserved by
full clique-sums.
\begin{corollary}\label{cor-csum}
Let $\GG$ be a class of graphs of chromatic number at most $k$. If $\GG'$ is the class
of graphs obtained from $\GG$ by repeatedly performing full clique-sums, then
$$\cbdim(\GG')\le \cbdim(\GG) + 2k.$$
\end{corollary}
\begin{proof}
Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
Without loss of generality, the labelling of the graphs is chosen so that we first
perform the full clique-sum on $G_1$ and $G_2$, then on the resulting graph and $G_3$, and so on.
Let $C^\star_1=\emptyset$ and for $i=2,\ldots,m$, let $C^\star_i$ be the root clique of $G_i$ on which it is
glued in the full clique-sum operation. By Lemmas~\ref{lem-ecbdim-cbdim} and \ref{lem-apex-cs},
$G_i$ has a $C_i^\star$-clique-sum extendable touching representation by comparable boxes in $\mathbb{R}^d$,
where $d=\cbdim(\GG) + 2k$. Repeatedly applying Lemma~\ref{lem-cs}, we conclude that
$\cbdim(G)\le d$.
\end{proof}
By Lemmas~\ref{lemma-chrom} and \ref{lemma-subg}, this gives the following bounds.
\begin{corollary}\label{cor-csump}
Let $\GG$ be a class of graphs of comparable box dimension at most $d$.
\begin{itemize}
\item The class $\GG'$ of graphs obtained from $\GG$ by repeatedly performing full clique-sums
has comparable box dimension at most $d + 2\cdot 3^d$.
\item The class of graphs obtained from $\GG$ by repeatedly performing clique-sums
has comparable box dimension at most $625^d$.
\end{itemize}
\end{corollary}
\begin{proof}
The former bound directly follows from Corllary~\ref{cor-csum} and the bound on the chromatic number
from Lemma~\ref{lemma-chrom}. For the latter one, we need to bound the star chromatic number of $\GG'$.
Suppose a graph $G$ is obtained from $G_1, \ldots, G_m\in\GG$ by performing full clique-sums.
For $i=1,\ldots, m$, suppose $G_i$ has an acyclic coloring $\varphi_i$ by at most $k$ colors.
Note that the vertices of any clique get pairwise different colors, and thus by permuting the colors,
we can ensure that when we perform the full clique-sum, the vertices that are identified have the same
color. Hence, we can define a coloring $\varphi$ of $G$ such that for each $i$, the restriction of
$\varphi$ to $V(G_i)$ is equal to $\varphi_i$. Let $C$ be the union of any two color classes of $\varphi$.
Then for each $i$, $G_i[C\cap V(G_i)]$ is a forest, and since $G[C]$ is obtained from these graphs
by full clique-sums, $G[C]$ is also a forest. Hence, $\varphi$ is an acyclic coloring of $G$
by at most $k$ colors. By~\cite{albertson2004coloring}, $G$ has a star coloring by at most $2k^2-k$ colors.
Hence, Lemma~\ref{lemma-chrom} implies that $\GG'$ has star chromatic number at most $2\cdot 25^d - 5^d$.
The bound on the comparable box dimension of subgraphs of graphs from $\GG'$ then follows from Lemma~\ref{lemma-subg}.
\end{proof}
\section{The strong product structure and minor-closed classes} \section{The strong product structure and minor-closed classes}
A \emph{$k$-tree} is any graph obtained by repeated full clique-sums on cliques of size $k$ from cliques of size at most $k+1$.
A \emph{$k$-tree-grid} is a strong product of a $k$-tree and a path.
An \emph{extended $k$-tree-grid} is a graph obtained from a $k$-tree-grid by adding at most $k$ apex vertices.
Dujmovi{\'c} et al.~\cite{DJM+} proved the following result. Dujmovi{\'c} et al.~\cite{DJM+} proved the following result.
\begin{theorem}\label{thm-prod} \begin{theorem}\label{thm-prod}
Any graph $G$ is a subgraph of the strong product of a $k$-tree, a path, and $K_m$, where Any graph $G$ is a subgraph of the strong product of a $k$-tree-grid and $K_m$, where
\begin{itemize} \begin{itemize}
\item $k=3$ and $m=3$ if $G$ is planar, and \item $k=3$ and $m=3$ if $G$ is planar, and
\item $k=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$. \item $k=4$ and $m=\max(2g,3)$ if $G$ has Euler genus at most $g$.
\end{itemize} \end{itemize}
Moreover, for every $t$, there exists a $k$ such that any Moreover, for every $t$, there exists an integer $k$ such that any
$K_t$-minor-free graph $G$ is a subgraph of a graph obtained from $K_t$-minor-free graph $G$ is a subgraph of a graph obtained by repeated clique-sums
successive clique-sums of graphs, that are obtained from the strong from extended $k$-tree-grids.
product of a path and a $k$-tree, by adding at most $k$ apex vertices.
\end{theorem} \end{theorem}
Let us first bound the comparable box dimension of a graph in terms of Let us first bound the comparable box dimension of a graph in terms of
its Euler genus. As paths and $m$-cliques admit touching its Euler genus. As paths and $m$-cliques admit touching
representations with hypercubes of unit size in $\mathbb{R}^{1}$ and representations with hypercubes of unit size in $\mathbb{R}^{1}$ and
in $\mathbb{R}^{\lceil \log_2 m \rceil}$ respectively, by in $\mathbb{R}^{\lceil \log_2 m \rceil}$ respectively, by
Lemma~\ref{lemma-sp} it suffice to bound the comparable box Lemma~\ref{lemma-sp} it suffices to bound the comparable box
dimension of $k$-trees. dimension of $k$-trees.
\begin{theorem}\label{thm-ktree} \begin{theorem}\label{thm-ktree}
For any $k$-tree $G$, $\cbdim(G) \le \ecbdim(G) \le k+1$. For any $k$-tree $G$, $\cbdim(G) \le \ecbdim(G) \le k+1$.
\end{theorem} \end{theorem}
\begin{proof} \begin{proof}
Note that there exists a $k$-tree $G'$ having a $k$-clique $C^\star$ Let $H$ be a complete graph with $k+1$ vertices and let $C^\star$ be
such that $G'\setminus V(C^\star)$ corresponds to $G$. Let us construct a clique of size $k$ in $H$. By Lemma~\ref{lem-cs}, it suffices
a $C^\star$-clique-sum extendable representation of $G'$ and note that to show that $H$ has a $C^\star$-clique-sum extendable touching representation
it induces a $\emptyset$-clique-sum extendable representation of by hypercubes in $\mathbb{R}^{k+1}$. Let $V(C^\star)=\{v_1,\ldots,v_k\}$.
$G$. We construct the representation $h$ so that (v1) holds with $d_{v_i}=i$ for each $i$;
this uniquely determines the hypercubes $h(v_1)$, \ldots, $h(v_k)$.
Note that $G'$ can be obtained by starting with a $(k+1)$-clique For the vertex $v_{k+1} \in V(H)\setminus V(C^\star)$, we set $h(v_{k+1})=[0,1/2]^{k+1}$.
containing $C^\star$, and by performing successive full clique-sums of This ensures that the \textbf{(vertices)} conditions holds.
$K_{k+1}$ on a $K_k$ subclique. By Lemma~\ref{lem-cs}, it suffice to
show that $K_{k+1}$, the $(k+1)$-clique with vertex set $\{v_1, For the \textbf{(cliques)} conditions, let us set
\ldots, v_{k+1}\}$, has a $(K_{k+1} -\{v_{k+1}\})$-clique-sum
extendable touching representation by hypercubes. Let us define such
touching representation $h$ as follows:
\begin{itemize}
\item $h(v_i)[i] = [-1,0] $ if $i\le k$
\item $h(v_i)[j] = [0,1] $ if $i\le k$ and $i\neq j$
\item $h(v_{k+1})[j] = [0,\frac12]$ for any $j$
\end{itemize}
One can easily check that the (vertices)
conditions are fulfilled. For the (cliques) conditions let us set
the point $p(C)$ for every clique $C$ as follows: the point $p(C)$ for every clique $C$ as follows:
\begin{itemize} \begin{itemize}
\item $p(C)[i] = 0 $ for every $i\le k$ and if $v_i\in C$ \item $p(C)[i] = 0 $ for every $i\le k$ such that $v_i\in C$
\item $p(C)[i] = \frac14 $ for every $i\le k$ and if $v_i\notin C$ \item $p(C)[i] = \frac14 $ for every $i\le k$ such that $v_i\notin C$
\item $p(C)[k+1] = \frac12 $ if $v_{k+1}\in C$ \item $p(C)[k+1] = \frac12 $ if $v_{k+1}\in C$
\item $p(C)[k+1] = \frac34 $ if $v_{k+1}\notin C$ \item $p(C)[k+1] = \frac34 $ if $v_{k+1}\notin C$
\end{itemize} \end{itemize}
By construction, it is clear that $p(C) \in h(v_i)$ if and only if By construction, it is clear that for each vertex $v\in V(H)$, $p(C) \in h(v)$ if and only if
$v_i\in V(C)$. Let us check the other (cliques) conditions. $v\in V(C)$.
For any two distinct cliques $C_1$ and $C_2$ the points $p(C_1)$ and For any two distinct cliques $C_1$ and $C_2$ the points $p(C_1)$ and
$p(C_2)$ are distinct. Indeed, if $|V(C_1)|\ge |V(C_2)|$ there is a $p(C_2)$ are distinct. Indeed, by symmetry we can assume that for some $i$,
vertex $v_i\in V(C_1)\setminus V(C_2)$, and this implies that we have $v_i\in V(C_1)\setminus V(C_2)$, and this implies that $p(C_1)[i] < p(C_2)[i]$.
$p(C_1)[i] < p(C_2)[i]$. Hence, the condition (v1) holds.
For a vertex $v_i$ and a clique $C$, the boxes $h(v_i)$ and Consider now a vertex $v_i$ and a clique $C$. As we observed before, if $v_i\not\in V(C)$,
$h^\varepsilon(C)$ intersect if and only if $v_i\in V(C)$. Indeed, if then $p(C) \not\in h(v_i)$, and thus $h^\varepsilon(C)$ and $h(v_i)$ are disjoint (for sufficiently small $\varepsilon>0$).
$v_i\in V(C)$ then $p(C)\in h(v_i)$ and $p(C)\in h^\varepsilon(C)$, and If $v_i\in C$, then the definitions ensure that $p(C)[i]$ is equal to the maximum of $h(v_i)[i]$,
if $v_i\notin V(C)$ then $h(v_i)[i] = [-1,0]$ if $i\le k$ and that for $j\neq i$, $p(C)[j]$ is in the interior of $h(v_i)[j]$, implying
(resp. $h(v_i)[i] = [0, \frac12]$ if $i= k+1$) and $h^\varepsilon(C)[i] = $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for sufficiently small $\varepsilon>0$.
[\frac14,\frac14+\varepsilon]$ (resp. $h^\varepsilon(C)[i] = \end{proof}
[\frac34,\frac34+\varepsilon]$). Finally, if $v_i\in V(C)$ we have that The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree.
$h(v_i)[i] \cap h^\varepsilon(C)[i] = \{p(C)[i]\}$ and that $h(v_i)[j] Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree}
\cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for any $j\neq i$ extends to graphs of treewidth at most $k$.
and any $\varepsilon <\frac14$. This concludes the proof of the theorem. \begin{corollary}\label{cor-tw}
Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$.
\end{corollary}
\begin{proof}
Let $k=\tw(G)$. Observe that there exists a $k$-tree $T$ with the root clique $C^\star$ such that $G\subseteq T-V(C^\star)$.
Inspection of the proof of Theorem~\ref{thm-ktree} (and Lemma~\ref{lem-cs}) shows that we obtain
a representation $h$ of $T-V(C^\star)$ in $\mathbb{R}^{k+1}$ such that
\begin{itemize}
\item the vertices are represented by hypercubes of pairwise different sizes,
\item if $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$, then $h(u)\cap h(v)$ is a facet of $h(u)$ incident
with its point with minimum coordinates, and
\item for each vertex $u$ and each facet of $h(u)$ incident with its point with minimum coordinates, there exists
at most one vertex $v$ such that $uv\in E(T-V(C^\star))$ and $h(u)\sqsubseteq h(v)$.
\end{itemize}
If for some $u,v\in V(G)$, we have $uv\in E(T)\setminus E(G)$, where without loss of generality $h(u)\sqsubseteq h(v)$,
we now alter the representation by shrinking $h(u)$ slighly away from $h(v)$ (so that all other touchings are preserved).
Since the hypercubes of $h$ have pairwise different sizes, the resulting touching representation of $G$ is by comparable boxes.
\end{proof} \end{proof}
Note that actually the bound on the comparable boxes dimension of Theorem~\ref{thm-ktree}
extends to graphs of treewidth $k$. For this, note that the construction in this proof can
provide us with a representation $h$ of any $k$-tree $G$ with hypercubes of distinct sizes.
Note also that this representation is such that for any two adjacent vertices $u$ and $v$,
with $h(u) \sqsubset h(v)$ say, the intersection $I = h(u) \cap h(v)$ is a facet of $h(u)$.
Actually $I[i] = h(u)[i]$ for every dimension, except one that we denote $j$. For this
dimension we have that $I[j]=\{c\}$ for some $c$, and that $h(u)[j]=[c,c+s]$,
where $s$ is the length of the sides of $h(u)$. In that context to delete an edge $uv$
one can simply replace $h(u)[j]=[c,c+s]$ with $[c+\varepsilon,c+s]$, for a sufficiently small $\varepsilon$.
One can proceed similarly for any subset of edges, and note that as the hypercubes in $h$ have
distinct sizes these small perturbations give rise to boxes that are still comparable.
Thus for any treewidth $k$ graph $H$ (that is a subgraph of a $k$-tree $G$) we have $\cbdim(H)\le k+1$.
As every planar graph $G$ has a touching representation by cubes in As every planar graph $G$ has a touching representation by cubes in
$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le $\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le 3$.
3$. For the graphs with higher Euler genus we can also derive upper For the graphs with higher Euler genus we can also derive upper
bounds. Indeed, combining the previous observation on the bounds. Indeed, combining the previous observation on the
representations of paths and $K_m$, with Theorem~\ref{thm-ktree}, representations of paths and $K_m$, with Theorem~\ref{thm-ktree},
Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain: Lemma~\ref{lemma-sp}, and Corollary~\ref{cor-subg} we obtain:
\begin{corollary}\label{cor-genus} \begin{corollary}\label{cor-genus}
For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$ For every graph $G$ of Euler genus $g$, there exists a supergraph $G'$
of $G$ such that $\cbdim(G')\le 5+1+\lceil \log_2 \max(2g,3)\rceil$. of $G$ such that $\cbdim(G')\le 6+\lceil \log_2 \max(2g,3)\rceil$.
Consequently, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 Consequently, $$\cbdim(G)\le 5\cdot 81^7 \cdot \max(2g,3)^{\log_2 81}.$$
81}.$$
\end{corollary} \end{corollary}
Similarly, we can deal with proper minor-closed classes.
Let us now finally prove Theorem~\ref{thm-minor}, using the structure \begin{proof}[Proof of Theorem~\ref{thm-minor}]
provided by Theorem~\ref{thm-prod}. We have seen that the strong Let $\GG$ be a proper minor-closed class. Since $\GG$ is proper, there exists $t$ such that $K_t\not\in \GG$.
product of a path and a $k$-tree has bounded comparable boxes 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
dimension, and by Lemma~\ref{lemma-apex} adding at most $k$ apex from extended $k$-tree-grids. As we have seen, $k$-tree-grids have comparable box dimension at most $k+2$,
vertices keeps the dimension bounded. Then by Lemma~\ref{lemma-chrom} and by Lemma~\ref{lemma-apex}, extended $k$-tree-grids have comparable box dimension at most $2k+2$.
and Lemma~\ref{lem-ecbdim-cbdim}, these graphs admit a By Corollary~\ref{cor-csump}, it follows that $\cbdim(\GG)\le 625^{2k+2}$.
$\emptyset$-clique-sum extendable representations in bounded \end{proof}
dimensions. As the obtained graphs have bounded dimension, by
Lemma~\ref{lemma-cliq} and Lemma~\ref