Commit 4c21c074 authored by Abhiruk Lahiri's avatar Abhiruk Lahiri
Browse files

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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