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

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

2004-08-19

arxiv.org/abs/math/0408262v2

Proc. Amer. Math. Soc. 134 (2006), no. 5, 1265--1270

Mathematics

Combinatorics

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

Scientific paper

In this paper we study implications of folds in both parameters of Lov\'asz' 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 $\phi$ is a closure operator on P, then $\Delta(P)$ collapses onto $\Delta(\phi(P))$.

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