Commit 4c21c074 by Abhiruk Lahiri

### a few minor typos

parent 382184bf
 ... ... @@ -34,7 +34,7 @@ %\nolinenumbers %uncomment to disable line numbering %\hideLIPIcs %uncomment to remove references to LIPIcs series (logo, DOI, ...), e.g. when preparing a pre-final version to be uploaded to arXiv or another public repository \hideLIPIcs %uncomment to remove references to LIPIcs series (logo, DOI, ...), e.g. when preparing a pre-final version to be uploaded to arXiv or another public repository %Editor-only macros:: begin (do not touch as author)%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \EventEditors{John Q. Open and Joan R. Access} ... ... @@ -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 ... ... @@ -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 ... ... @@ -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. \end{proof} \end{document} \end{document} \ No newline at end of file
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