A simple proof for folds on both sides in complexes of graph homomorphisms

Abstract

In this paper we study implications of folds in both parameters of Lovász' Hom(-,-) complexes. There is an important connection between the topological properties of these complexes and lower bounds for chromatic numbers. We give a very short and conceptual proof of the fact that if G-v is a fold of G, then Bd(Hom(G,H)) collapses onto Bd Hom(G-v,H), whereas Hom(H,G) collapses onto Hom(H,G-v). We also give an easy inductive proof of the only nonelementary fact which we use for our arguments: if $\varphi$ is a closure operator on P, then $\Delta (P)$ collapses onto $\Delta (\varphi (P))$.

Final version, to appear in Proceedings of the American Mathematical Society

References in corpus (1)

• Complexes of graph homomorphisms