From 53a740d38e9a43b82eeca8edf262ae00586ee809 Mon Sep 17 00:00:00 2001 From: =?UTF-8?q?Daniel=20Gon=C3=A7alves?= <daniel.goncalves@lirmm.fr> Date: Fri, 11 Mar 2022 02:47:21 +0100 Subject: [PATCH] Update comparable-box-dimension.tex --- comparable-box-dimension.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/comparable-box-dimension.tex b/comparable-box-dimension.tex index ae145ab..e07bda1 100644 --- a/comparable-box-dimension.tex +++ b/comparable-box-dimension.tex @@ -436,7 +436,7 @@ behave well with respect to full clique-sums. sum extendable representation $h$ by comparable boxes in $\mathbb{R}^{\max(d_1,d_2)}$. \end{lemma} -The proof is in the appendix, but the idea is to translate (allowing +The proof is omitted, but the idea is to translate (allowing also exchanges of dimensions) and scale $h_2$ to fit in $h_1^\varepsilon(C_1)$. The following lemma enables us to pick the root clique at the expense of increasing @@ -446,7 +446,7 @@ the dimension by $\omega(G)$. $C^\star$-clique-sum extendable touching representation by comparable boxes in $\mathbb{R}^d$, for $d = |V(C^\star)| + \ecbdim(G\setminus V(C^\star))$. \end{lemma} - The proof is also in the appendix, but it essentially the same as the one of + The proof is also omitted, but it is essentially the same as the one of Lemma~\ref{lemma-apex}. The following lemma provides an upper bound on $\ecbdim(G)$ in terms of $\cbdim(G)$ and $\chi(G)$. @@ -618,7 +618,7 @@ $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for suffici \end{proof} The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree. Note that actually the bound on the comparable box dimension of Theorem~\ref{thm-ktree} -extends to graphs of treewidth at most $k$ (see the proof in the appendix). +extends to graphs of treewidth at most $k$ (proof omitted). \begin{corollary}\label{cor-tw} Every graph $G$ satisfies $\cbdim(G)\le\tw(G)+1$. \end{corollary} @@ -815,4 +815,4 @@ has a sublinear separator of size $O_{t,s,d}\bigl(|V(G)|^{\tfrac{d}{d+1}}\bigr). \end{corollary} \bibliography{data} -\end{document} \ No newline at end of file +\end{document} -- GitLab