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

Dmitry N. Kozlov
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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 $ϕ$ is a closure operator on P, then $Δ(P)$ collapses onto $Δ(ϕ(P))$.

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Dmitry N. Kozlov. 2004-12-03. A simple proof for folds on both sides in complexes of graph homomorphisms. https://arxiv.org/abs/math/0408262

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