Box complexes, neighborhood complexes, and the chromatic number

2003/10/21 by Peter Csorba, Carsten Lange, Ingo Schurr +1 · 1 citation
Mathematics · #math.CO #math.AT #msc:05C15 #msc:55P10 #msc:57M15

paper · pdf

published as Journal of Combinatorial Theory, Series A 108 (2004), pp. 159-168. · 8 pages, 1 figure

arxiv created 2003/10/21 · arxiv updated 2009/12/01

Abstract

Lovasz's striking proof of Kneser's conjecture from 1978 using the Borsuk--Ulam theorem provides a lower bound on the chromatic number of a graph. We introduce the shore subdivision of simplicial complexes and use it to show an upper bound to this topological lower bound and to construct a strong Z₂-deformation retraction from the box complex (in the version introduced by Matousek and Ziegler) to the Lovasz complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several ``intermediate'' complexes.

Cited by

• Covering and labeling generalizations of the Borsuk-Ulam theorem

Related

• Box complexes: at the crossroad of graph theory and topology
• Topological lower bounds for the chromatic number: A hierarchy
• Chromatic number, Clique number, and Lovász's bound: In a comparison
• Chromatic numbers, morphism complexes, and Stiefel-Whitney characteristic classes
• Chromatic Numbers of Simplicial Manifolds