A touching representation of axis-aligned boxes in $\mathbb{R}^d$ is said \emph{fully touching} if any two intersecting boxes intersect on a $(d-1)$-dimensional box. Note that the construction above is fully touching.
A touching representation of axis-aligned boxes in $\mathbb{R}^d$ is said to be \emph{fully touching} if any two intersecting boxes intersect on a $(d-1)$-dimensional box. Note that the construction above is fully touching.
Indeed, two intersecting boxes corresponding to vertices $u,v$of colors $c(u) < c(v)$, only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} leads to the following.
Indeed, two intersecting boxes corresponding to vertices $u,v$with colors $c(u) < c(v)$ only touch at coordinate $2/5$ in the $(d+c(u))^\text{th}$ dimension, while they fully intersect in every other dimension. This observation with Lemma~\ref{lemma-chrom} leads to the following.
\begin{corollary}
\begin{corollary}
\label{cor-fully-touching}
\label{cor-fully-touching}
Any graph $G$ has a fully touching representation of comparable axis-aligned boxes in $\mathbb{R}^d$, where $d=\cbdim(G)+3^{\cbdim(G)}$.
Any graph $G$ has a fully touching representation of comparable axis-aligned boxes in $\mathbb{R}^d$, where $d=\cbdim(G)+3^{\cbdim(G)}$.
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@@ -807,10 +807,10 @@ Since the hypercubes of $h$ have pairwise different sizes, the resulting touchin
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@@ -807,10 +807,10 @@ Since the hypercubes of $h$ have pairwise different sizes, the resulting touchin
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 $\cbdim(G)\le3$.
$\mathbb{R}^3$~\cite{felsner2011contact}, we have that $\cbdim(G)\le3$.
For the graphs with higher Euler genus we can also derive upper
For 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 $G'$
