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\EventEditors{John Q. Open and Joan R. Access}

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@@ -147,7 +147,7 @@ For any graph $G$, we have $\omega(G)\le 2^{\cbdim(G)}$.

We may assume that $G$ has bounded comparable box dimension

witnessed by a representation $f$. To represent any clique $A =\{a_1,\ldots,a_w\}$ in $G$, the

corresponding boxes $f(a_1),\ldots,f(a_w)$ have pairwise non-empty

intersection. Since axis-aligned boxes have the Helly property, there

intersections. Since axis-aligned boxes have the Helly property, there

is a point $p \in\mathbb{R}^d$ contained in $f(a_1)\cap\cdots\cap

f(a_w)$. As each box is full-dimensional, its interior intersects at

least one of the $2^d$ orthants at $p$. At the same time, it follows from the definition

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@@ -217,7 +217,7 @@ is at most $(3^d-1)^2$. This means that when choosing the color of $v_i$ greedil

\section{Operations}

It is clear that given a touching representation of a graph $G$, one

easily obtains a touching representation by boxes of an induced

can 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. In the following, to describe

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@@ -1018,4 +1018,4 @@ we now alter the representation by shrinking $h(u)$ slightly away from $h(v)$ (s

Since the hypercubes of $h$ have pairwise different sizes, the resulting touching representation of $G$ is by comparable boxes.