Subcomplexes of box complexes of graphs

Abstract

The box complex ${\sf B}(G)$ of a graph $G$ is a simplicial $\mathbb{Z}_2$-complex defined by J. Matoušek and G.M. Ziegler in \cite{MZ04}. They proved that $\chi (G)\geq \text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G)\| )+2$, where $\chi (G)$ is the chromatic number of $G$ and $\text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G)\| )$ is the $\mathbb{Z}_2$-index of ${\sf B}(G)$. In this paper, to study topology of box complexes, for the union $G\cup H$ of two graphs $G$ and $H$, we compare ${\sf B}(G\cup H)$ with its subcomplex ${\sf B}(G)\cup {\sf B}(H)$. We give a sufficient condition on $G$ and $H$ so that ${\sf B}(G\cup H)={\sf B}(G)\cup {\sf B}(H)$ and ${\sf B}(G\cap H)={\sf B}(G)\cap {\sf B}(H)$ hold. Moreover, under that condition, we show

\max \{\chi (G), \chi (H)\}\leq \chi (G\cup H)\leq \max \{\chi (G)+l_H, \chi (H)\},

where $l_H$ is the number defined in Definition 3.8. Also we prove

\text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G\cup H)\| )=\max \{\,\text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G)\| ),\, \text{ind}_{\mathbb{Z}_2}(\| {\sf B}(H)\| )\,\}

if $\max \{\,\text{ind}_{\mathbb{Z}_2}(\| {\sf B}(G)\| ),\,\text{ind}_{\mathbb{Z}_2}(\| {\sf B}(H)\| )\,\}\geq 1$.

The complex ${\sf B}(G)$ of a graph $G$ contains a 1-dimensional free $\mathbb{Z}_2$-subcomplex $\overline{G}$ of ${\sf B}(G)$, defined in \cite{Aki07}. As a supplement to \cite{Aki07}, we show that for a connected graph $G$, ${\sf B}(G)$ is disconnected if and only if $\overline{G}$ is disconnected if and only if $G$ contains no cycles of odd length, or equivalently, $G$ is bipartite.