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’ H o m ( − , − ) \mathtt {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 G-v is a fold of G G , then bd ⁡ H o m ( G , H ) \operatorname {bd}\mathtt {Hom}(G,H) collapses onto bd ⁡ H o m ( G − v , H ) \operatorname {bd}\mathtt {Hom}(G-v,H) , whereas H o m ( H , G ) \mathtt {Hom}(H,G) collapses onto H o m ( H , G − v ) \mathtt {Hom}(H,G-v) . We also give an easy inductive proof of the only nonelementary fact which we use for our arguments: if φ \varphi is a closure operator on P P , then Δ ( P ) \Delta (P) collapses onto Δ ( φ ( P ) ) \Delta (\varphi (P)) .