diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex
index a2b9669d4f40be6763e1be70cde0c4314fe86cb1..f8feea45f49892de3d4faf2c689d4d460640fa66 100644
--- a/comparable-box-dimension.tex
+++ b/comparable-box-dimension.tex
@@ -162,7 +162,7 @@ vertices. We proceed similarly to bound the chromatic number.
\section{Operations}
It is clear that given a touching representation of a graph $G$, one
-easily obtains a touching representation with boxes of an induced
+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.
@@ -189,8 +189,8 @@ Then $h$ is a touching representation of $G$ by comparable boxes in $\mathbb{R}^
Let $G\boxtimes H$ denote the \emph{strong product} of the graphs $G$
and $H$, i.e., the graph with vertex set $V(G)\times V(H)$ and with
-distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if
-either $u_1=u_2$ or $u_1u_2\in E(G)$ and either $v_1=v_2$ or
+distinct vertices $(u_1,v_1)$ and $(u_2,v_2)$ adjacent if and only if,
+either $u_1=u_2$ or $u_1u_2\in E(G)$, and either $v_1=v_2$ or
$v_1v_2\in E(G)$. To obtain a touching representation of $G\boxtimes
H$ it suffice to take a product of representations of $G$ and $H$, but
the obtained representation may contain uncomparable boxes. Thus,
@@ -208,7 +208,7 @@ overcome this issue, by constraining one of the representations.
The proof simply consists in taking a product of the two
representations. Indeed, consider a touching respresentation with
comparable boxes $g$ of $G$ in $\mathbb{R}^{d_G}$, with
- $d_G=\cbdim(G)$, and the depresentation $h$ of $H$. Let us define a
+ $d_G=\cbdim(G)$, and the representation $h$ of $H$. Let us define a
representation $f$ of $G\boxtimes H$ in $\mathbb{R}^{d_G+d_H}$ as
follows.
$$f((u,v))[i]=\begin{cases}
@@ -221,7 +221,7 @@ overcome this issue, by constraining one of the representations.
Now let us observe that for any two vertices $u, u'$ of $G$, there
is an hyperplane separating the interiors of $g(u)$ and $g(u')$, and
similarly for $h$ and $H$. This implies that the boxes in $f$ are
- interior disjoint. Indeed, the same hyperplane that separates $g(u)$
+ interiorly disjoint. Indeed, the same hyperplane that separates $g(u)$
and $g(u')$, when extended to $\mathbb{R}^{d_G+d_H}$, now separates
any two boxes $f((u,v))$ and $f((u',v'))$. This implies that $f$ is
a touching representation of a subgraph of $G\boxtimes H$.
@@ -231,7 +231,7 @@ overcome this issue, by constraining one of the representations.
a point $p_G$ in the intersection of $g(u)$ and $g(u')$, and a point
$p_H$ in the intersection of $h(v)$ and $h(v')$. Indeed, one can
obtain $p_G$ and $p_H$ by projecting $p$ in $\mathbb{R}^{d_G}$ or in
- $\mathbb{R}^{d_G}$ respectively, and conversely $p$ can be obtained
+ $\mathbb{R}^{d_H}$ respectively, and conversely $p$ can be obtained
by taking the product of $p_G$ and $p_H$. Thus $f$ is indeed a
touching representation of $G\boxtimes H$.
\end{proof}
@@ -247,10 +247,6 @@ 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.
-
-
-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}
@@ -340,7 +336,7 @@ boxes or not) $h$ of $G$ in $\mathbb{R}^d$ is called
Note that we may consider that the root clique is empty, that is the
empty subgraph with no vertices. In that case the clique is denoted
$\emptyset$. Let $\ecbdim(G)$ be the minimum dimension such that $G$
-has a $\emptyset$-clique-sum extendable touching representation with
+has a $\emptyset$-clique-sum extendable touching representation by
comparable boxes. The following lemma ensures that clique-sum
extendable representations behave well with respect to full
clique-sums.
@@ -352,12 +348,8 @@ clique-sums.
respectively. Let $G$ be the graph obtained after performing a full
clique-sum of these two graphs on any clique $C_1$ of $G_1$, and on
the root clique $C^*_2$ of $G_2$. Then $G$ admits a $C^*_1$-clique
- sum extendable representation with comparable boxes $h$ in
- $\mathbb{R}^{\max(d_1,d_2)}$. Furthermore, we have that the aspect
- ratio of $h(v)$ is the same as the one of $h_1(v)$, if $v\in
- V(G_1)$, or the same as $h_2(v)$ if $v\in V(G_2)\setminus V(C^*_2)$.
- \note{ shall we remove the aspect ratio thing ? only needed for the
- hypercubes of the k-trees, but those are not really needed...}
+ sum extendable representation by comparable boxes $h$ in
+ $\mathbb{R}^{\max(d_1,d_2)}$.
\end{lemma}
\begin{proof}
The idea is to translate (allowing also exchanges of dimensions) and
@@ -394,26 +386,25 @@ clique-sums.
by $\epsilon$. More formally, for any vertex $v\in V(G_2) \setminus
\{v_1,\ldots,v_k\}$, we set $h(v)[\sigma(i)] = p(C_1)[\sigma(i)] +
\epsilon h_2(v)[i]$ for $i\in \{1 ,\ldots,d_2\}$, and $h(v)[j] =
- p(C_1)[j] + [0, \epsilon/2]$, otherwise (for any $j$ not in the
+ [p(C_1)[j], p(C_1)[j] +\epsilon/2]$, otherwise (for any $j$ not in the
image of $\sigma$). Note that if we apply the same mapping from
- $h_2$ to $h$, to the vertices of $\{v_1,\ldots,v_k\}$, then the
- image of $h_2(v_i)$ fit inside the (previously defined) box
+ $h_2$ to $h$, to the boxes $h_2(v_i)$ for $i\in \{1,\ldots,k\}$, then the
+ image of $h_2(v_i)$ fits inside the (previously defined) box
$h(v_i)$. Similarly the clique points $p_2(C)$ become $p(C)$ by
setting $p(C)[\sigma(i)] = p(C_1)[\sigma(i)] + \epsilon p_2(C)[i]$
- for $i\in \{1 ,\ldots,d_2\}$, and $p(C)[j] = p(C_1)[j] + [0,
- \frac14]$, otherwise.
+ for $i\in \{1 ,\ldots,d_2\}$, and $p(C)[j] = p(C_1)[j] + \epsilon/4$, otherwise.
Note that we have defined (differently) both $h^\epsilon(C_1)$
(resp. $p(C_1)$) and $h^\epsilon(C^*_2)$ (resp. $p(C^*_2)$), despite
the fact that those cliques were merged. In the following we use
$h^\epsilon(C_1)$ and $p(C_1)$ only for the purpose of the
- proof. The point and the box corresponding to these clique in $h$ is
+ proof. The point and the box corresponding to this clique in $h$ is
$p(C^*_2)$ and $h^\epsilon(C^*_2)$.
Let us now check that $h$ is a $C^*_1$-clique sum extendable
- representation with comparable boxes. The fact that the boxes are
+ representation by comparable boxes. The fact that the boxes are
comparable follows from the fact that those of $V(G_1)$
- (resp. $V(G_1)$) are comparable in $h_1$ (resp. $h_2$) with the
+ (resp. $V(G_2)$) are comparable in $h_1$ (resp. $h_2$) with the
boxes of $V(C^*_1)$ (resp. $V(C^*_2)$) being hypercubes of side one,
and the other boxes being smaller. Clearly, by construction both
$h_1(u) \sqsubseteq h_1(v)$ or $h_2(u) \sqsubseteq h_2(v)$, imply
@@ -423,19 +414,27 @@ clique-sums.
We now check that $h$ is a contact representation of $G$. For $u,v
\in V(G_1)$ (resp. $u,v \in V(G_2) \setminus \{v_1,\ldots,v_k\}$) it
- is clear that $h(u)$ and $h(v)$ are interior disjoint, and that they
+ is clear that $h(u)$ and $h(v)$ have disjoint interiors, and that they
intersect if and only if $h_1(u)$ and $h_1(v)$ intersect (resp. if
$h_2(u)$ and $h_2(v)$ intersect). Consider now a vertex $u \in
V(G_1)$ and a vertex $v \in V(G_2) \setminus \{v_1,\ldots,v_k\}$. As
$h(v)$ fits inside $h^\epsilon(C_1)$, we have that $h(u)$ and $h(v)$
- are interior disjoint. Furthermore, if they intersect then $u\in
+ have disjoint interiors. Furthermore, if they intersect then $u\in
V(C_1)$, say $u=v_1$, and $h(v)[1] = [p(C_1)[1], p(C_1)[1]+\alpha]$
for some $\alpha>0$. By construction, this implies that $h_2(v_1)$
and $h_2(v)$ intersect.
Finally for the $C^*_1$-clique-sum extendability, one can easily
- check that the (vertex) conditions hold, such as the (clique)
- conditions.
+ check that the (vertices) conditions hold. For the (cliques)
+ conditions, as the mapping from $p_2(C)$ to $p(C)$ (extended to a
+ mapping from $\mathbb{R}^{d_2}$ to $\mathbb{R}^{d}$) is injective,
+ we have that (c1) clearly holds. For (c2) one has to notice that if
+ $d=d_2$, then the mapping from $h_2$ to $h$ extended to the clique boxes
+ would lead to the same clique boxes $h^{\epsilon'}(C)$, with the same point $p(C)$ in their lower corner.
+ If there are extra dimensions, that is if $d> d_2$, then for any such
+ dimension $j$ that is not in the image of $\sigma$, we have that
+ $h^{\epsilon'}(C)[j] = [p(C_1)[j] + \epsilon/4, p(C_1)[j] + \epsilon/4 + \epsilon']
+ \subset [p(C_1)[j], p(C_1)[j] + \epsilon/2] = h(v)[j]$.
\end{proof}
The following lemma shows that any graphs has a $C^*$-clique-sum
@@ -443,7 +442,7 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
\ecbdim(G)$ and for any clique $C^*$.
\begin{lemma}\label{lem-apex-cs}
For any graph $G$ and any clique $C^*$, we have that $G$ admits a
- $C^*$-clique-sum extendable touching representation with comparabe
+ $C^*$-clique-sum extendable touching representation by comparabe
boxes in $\mathbb{R}^d$, for $d = |V(C^*)| + \ecbdim(G\setminus
V(C^*))$.
\end{lemma}
@@ -458,12 +457,40 @@ extendable representation in $\mathbb{R}^d$, for $d= \omega(G) +
$d_{v_i} = i$. For each vertex $u\in V(G)\setminus V(C^*)$, 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] =
- h'(u)[i-k]$. We proceed similarly for the clique points. For any
+ \alpha_i h'(u)[i-k]$, for some $\alpha_i>0$. The values $\alpha_i>0$
+ are chosen suffciently small so that $h(u)[i] \subset [0,1)$, whenever $u\notin V(C^*)$.
+ 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
V(C)$, and $p(C)[i] = 1/4$ if $v_i \notin V(C)$. For $i>k$ we have
to refer to the clique point $p'(C')$ of $C'=C\setminus
- \{v_1,\ldots,v_k\}$, as we set $p(C)[i] = p'(C')[i-k]$. One can
- easily check that $h$ is as desired.
+ \{v_1,\ldots,v_k\}$, as we set $p(C)[i] = \alpha_i p'(C')[i-k]$.
+
+ 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^*)$ and every
+ $v_i \in V(C^*)$, we have that $h$ is a touching representation by comparable boxes.
+ By the construction, it is clear that $h$ is a representation of $G$.
+ For the $C^*$-clique-sum extendability, it is clear that the (vertices) conditions hold.
+ For the (cliques) condition (c1), let us first consider two 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^*)$. 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]$.
+ Otherwise, if $C'_1 \neq C'_2$, we have that $p'(C'_1) \neq p'(C'_2)$, which leads to
+ $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^*)$ and a clique $C$ of $G$ containing $v$.
+ In the first dimensions $i \le k$, we always have $h^\epsilon(C)[i] \subseteq h(v)[i]$. Indeed, if $v_i \in V(C)$ we have
+ $h^\epsilon(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)$
+ we have $h^\epsilon(C)[i] \subseteq [1/4,1/2] \subseteq h(v)[i]$. Then for the last $d'$ dimensions, by definition of $h'$,
+ we have that $h^\epsilon(C)[i] \subseteq h(v)[i]$ for every $i>k$, except one,
+ for which $h^\epsilon(C)[i] \cap h(v)[i] = \{p(C)[i]\}$. This completes the first case
+ and we now consider a vertex $v\in V(G)\setminus V(C^*)$ and a clique $C$ of $G$ not containing $v$.
+ As $v\notin V(C')$, there is an hyperplane
+ ${\mathcal H}' = \{ p\in \mathbb{R}^{d'}\ |\ p[i] = c\}$ that separates $p'(C')$ and $h'(v)$.
+ 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)$.
+ Now we consider a vertex $v_i \in V(C^*)$, and we note that for any clique $C$ containing $v_i$
+ we have that $h^\epsilon(C)[i] \cap h(v_i)[i] = [0,\epsilon]\cap [-1,0] = \{0\}$, and $h^\epsilon(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
+ $h^\epsilon(C)[i] \cap h(v_i)[i] \subset (0,1)\cap [-1,0] = \emptyset$.
+ Condition (c2) is thus fulfilled and this completes the proof of the lemma.
\end{proof}
The following lemma provides an upper bound on $\ecbdim(G)$ in terms
@@ -472,7 +499,7 @@ of $\cbdim(G)$ and $\chi(G)$.
For any graph $G$, $\ecbdim(G) \le \cbdim(G) + \chi(G)$.
\end{lemma}
\begin{proof}
- Let $h$ be a touching representation with comparable boxes of $G$ in
+ Let $h$ be a touching representation by comparable boxes of $G$ in
$\mathbb{R}^d$, with $d=\cbdim(G)$, and let $c$ be a
$\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
@@ -495,20 +522,36 @@ of $\cbdim(G)$ and $\chi(G)$.
[0,2/5]&\text{ if $c(u) = i-d$}\\
[2/5,4/5]&\text{ otherwise (if $c(u) > i-d > 0$)}
\end{cases}$$
- For any clique $C'$ of $G$, let us denote $c(C)$, the set
+ For any clique $C'$ of $G$, let us denote $c(C')$, the color set
$\{c(u)\ |\ u\in V(C')\}$, and let $C$ be one of the maximal cliques
- containing $C'$.We now set
+ containing $C'$. We now set
$$p_2(C')[i]=\begin{cases}
p_1(C) &\text{ if $i\le d$}\\
2/5 &\text{ if $i-d \in c(C')$}\\%\text{if $\exists u\in V(C)$ with $c(u) = i-d$}\\
1/2 &\text{ otherwise}
\end{cases}
- $$ One can now check that this is a $\emptyset$-clique-sum
- extendable touching representation with comparable boxes. In particular,
- one should notice that for a vertex $u$ of a clique $C'$ we have that
- $h_2^{\epsilon}(C')[i] \subset h_2(u)[i]$ for every dimension $i$,
- except if $c(u)=i-d$. If $c(u)=i-d$, then $h_2(u)[i] \cap
- h_2^{\epsilon}(C')[i] = \{2/5\}$.
+ $$
+ As $h_2$ is an extension of $h_1$, and as for all the extra dimensions $j$ ($j>d$)
+ we have that $2/5 \in h_2(v)[j]$ for every vertex $v$, we have that $h_2$ is an
+ intersection representation of $G$. To prove that it is touching consider two adjacent
+ vertices $u$ and $v$ such that $c(u)<c(v)$, and let us note that $h_2(u)[d+c(u)] = [0,2/5]$
+ and $h_2(v)[d+c(u)] = [2/5,4/5]$. By construction, the boxes of $h_1$ are comparable boxes,
+ 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
+ comparables boxes.
+ For the $\emptyset$-clique-sum extendability, the (vertices) conditions clearly hold.
+ 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 $C_2$,
+ which clique points $p_2(C_1)$ and $p_2(C_2)$ are based on distinct maximum cliques, necessarily lead to distinct points.
+ In the case that $C_1$ and $C_2$ belong to some maximal clique $C$, we have that $c(C_1) \neq c(C_2)$
+ and this implies by construction that $p_2(C_1)$ and $p_2(C_2)$ are distinct. Thus (c1) holds.
+ By construction of $h_1$, we have that if $h_2^{\epsilon}(C')[i] \cap h_2(v)[i]$ is non-empty for every $i\le d$,
+ then we have that $h_2^{\epsilon}(C')[i] \subset h_2(v)[i]$ for every $i\le d$,
+ and we have that $v$ belongs to some maximal clique $C$ containing $C'$. If $v\notin V(C')$ note that
+ $p_2(C')[d+c(v)] = 1/2 \notin[0,2/5]=h_2(v)[d+c(v)]$, while if $v\in V(C')$ we have that
+ $h_2^{\epsilon}(C')[i] \subset [2/5,1/2+\epsilon] \subset h_2(v)[i]$ for every dimension $i>d$,
+ except if $c(v)=i-d$, and in that case $h_2(v)[i] \cap h_2^{\epsilon}(C')[i] =
+ [0,2/5]\cap[2/5,2/5+\epsilon] = \{2/5\}$. We thus have that (c2) holds, and this concludes the proof of the lemma.
\end{proof}
@@ -531,7 +574,7 @@ product of a path and a $k$-tree, by adding at most $k$ apex vertices.
\end{theorem}
Let us first bound the comparable box dimension of a graph in terms of
-its Euler genus. As paths and $m$-clique admit touching
+its Euler genus. As paths and $m$-cliques admit touching
representations with hypercubes of unit size in $\mathbb{R}^{1}$ and
in $\mathbb{R}^{\lceil \log_2 m \rceil}$ respectively, by
Lemma~\ref{lemma-sp} it suffice to bound the comparable box
@@ -552,7 +595,7 @@ containing $C^*$, and by performing successive full clique-sums of
$K_{k+1}$ 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 $\{v_1,
\ldots, v_{k+1}\}$, has a $(K_{k+1} -\{v_{k+1}\})$-clique-sum
-extendable touching representation with hypercubes. Let us define such
+extendable touching representation by hypercubes. Let us define such
touching representation $h$ as follows:
\begin{itemize}
\item $h(v_i)[i] = [-1,0] $ if $i\le k$
@@ -580,15 +623,28 @@ For a vertex $v_i$ and a clique $C$, the boxes $h(v_i)$ and
$h^\epsilon(C)$ intersect if and only if $v_i\in V(C)$. Indeed, if
$v_i\in V(C)$ then $p(C)\in h(v_i)$ and $p(C)\in h^\epsilon(C)$, and
if $v_i\notin V(C)$ then $h(v_i)[i] = [-1,0]$ if $i\le k$
-(resp. $h(v_i)[i] = [0 \frac12]$ if $i= k+1$) and $h^\epsilon(C)[i] =
+(resp. $h(v_i)[i] = [0, \frac12]$ if $i= k+1$) and $h^\epsilon(C)[i] =
[\frac14,\frac14+\epsilon]$ (resp. $h^\epsilon(C)[i] =
[\frac34,\frac34+\epsilon]$). Finally, if $v_i\in V(C)$ we have that
$h(v_i)[i] \cap h^\epsilon(C)[i] = \{p(C)[i]\}$ and that $h(v_i)[j]
\cap h^\epsilon(C)[j] = [p(C)[j],p(C)[j]+\epsilon]$ for any $j\neq i$
and any $\epsilon <\frac14$. This concludes the proof of the theorem.
\end{proof}
-
-As every planar graph $G$ has a touching representation with cubes in
+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 $[c+\epsilon,c+s]$, for a sufficiently small $\epsilon$.
+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
$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le
3$. For the graphs with higher Euler genus we can also derive upper
bounds. Indeed, combining the previous observation on the