Subcomplexes of box complexes of graphs (Q1032693)

The box complex \({\mathbf B}(G)\) of a graph \(G\) is a simplicial \({\mathbf Z}_2\)-complex defined by \textit{J. Matoušek} and \textit{G. M. Ziegler} in [``Topological lower bounds for the chromatic number: a hierarchy'', Jahresber. Dtsch. Math.-Ver. 106, No.~2, 71--90 (2004; Zbl 1063.05047)]. They proved that \(\chi(G)>\text{ind}_{{\mathbf Z}_2}(\|{\mathbf B}(G)\|)+2\), where \(\chi(G)\) is the chromatic number of \(G\) and \(\text{ind}_{{\mathbf Z}_2}({\mathbf B}(G)\|)\) is the \({\mathbf Z}_2\)-index of \({\mathbf B}(G)\). In this paper, to study topology of box complexes, for the union \(G\cup H\) of two graphs \(G\) and \(H\), we compare \(B(G\cup H)\) with its subcomplex \({\mathbf B}(G)\cup {\mathbf B}(H)\). We give a sufficient condition on \(G\) and \(H\) so that \({\mathbf B}(G\cup H) = {\mathbf B}(G)\cup {\mathbf B}(H)\) and \({\mathbf B}(G\cap H) = {\mathbf B}(G)\cap {\mathbf B}(H)\) 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 \[ \text{ind}_{{\mathbf Z}_2}(\|{\mathbf B}(G\cup H\|)=\max\{\text{ind}_{{\mathbf Z}_2}(\|{\mathbf B}G)\|),\text{ind}{{\mathbf Z}_2}, (\|{\mathbf B}(H)\|)\} \] if \(\max\{\text{ind}{{\mathbf Z}_2}(\|{\mathbf B}(G)\|),\text{ind}{{\mathbf Z}_2}(\|{\mathbf B}(H)\|)\} > 1\). The complex \({\mathbf B}(G)\) of a graph \(G\) contains a 1-dimensional free \({\mathbf Z}_2\)-subcomplex \(G\) of \({\mathbf B}(G)\), defined in [\textit{A. Kamibeppu}, ``Homotopy type of the box complexes of graphs without 4-cycles'', Tsukuba J. Math. 32, No.~2, 307--314 (2008; Zbl 1187.05032)]. As a supplement to the cited paper, we show that for a connected graph \(G\), \({\mathbf B}(G)\) is disconnected if and only if \(G\) is disconnected if and only if \(G\) contains no cycles of odd length, or equivalently, \(G\) is bipartite.