Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi

Abstract

The neighborhood complex N (G) of a graph G were introduced by L. Lovász in his proof of Kneser conjecture. He proved that for any graph G, χ(G) ≥ conn(N (G)) + 3.(1)In this article we show that for a class of exponential graphs the bound given in (1) is tight. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy. We were also able to find a class of exponential graphs, which are homotopy test graphs.Hedetniemi's conjecture states that the chromatic number of the categorical product of two graphs is the minimum of the chromatic number of the factors. Let M (G) denotes the Mycielskian of a graph G. We show that, for any graph G containing M (M (K n )) as a subgraph and for any graph H, if χ(G × H) = n + 1, then min{χ(G), χ(H)} = n + 1. Therefore, we enrich the family of graphs satisfying the Hedetniemi's conjecture.