Connectedness of certain graph coloring complexes

In this article, we consider the bipartite graphs K-2 x K-n . We prove that the connectedness of the complex Hom(K-2 x K-n, K-m) is m - n - 1 if m >= n and m -3 in all the other cases. Therefore, we show that for this class of graphs, Hom(G, K-m) is exactly (m - d - 2)-connected, m >= n, where d is the maximal degree of the graph G . (c) 2024 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.