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In this paper we study implications of folds in both parameters of Lovasz' 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 bdHom(G, H) collapses onto bdHom(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 φ is a closure operator on P, then A(P) collapses onto Δ(φ(P)).

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