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

Author: Dmitry N. Kozlov
Journal: Proc. Amer. Math. Soc. 134 (2006), 1265-1270
MSC (2000): Primary 05C15; Secondary 57M15
Published electronically: October 6, 2005
MathSciNet review: 2199168
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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 $ \varphi$ is a closure operator on $ P$, then $ \Delta(P)$ collapses onto $ \Delta(\varphi(P))$.

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Dmitry N. Kozlov
Affiliation: Department of Computer Science, Eidgenössische Technische Hochschule, Zürich, Switzerland

Keywords: Graphs, graph homomorphisms, \text{\tt{Hom}} complex, closure operator, collapse, fold, order complex, discrete Morse theory, graph coloring
Received by editor(s): September 1, 2004
Received by editor(s) in revised form: December 2, 2004
Published electronically: October 6, 2005
Additional Notes: This research was supported by Swiss National Science Foundation Grant PP002-102738/1
Communicated by: Paul Goerss
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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