Box complexes, neighborhood complexes, and the chromatic number

Authors:
Péter Csorba;Carsten Lange;Ingo Schurr;Arnold Wassmer
Affiliations:
Institute of Theoretical Computer Science, ETH Zürich, CH-8092 Zürich, Switzerland;Institute of Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany;Institute of Theoretical Computer Science, ETH Zürich, CH-8092 Zürich, Switzerland;Institute of Mathematics, MA 6-2, TU Berlin, D-10623 Berlin, Germany
Venue:
Journal of Combinatorial Theory Series A
Year:
2004

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Abstract

Lovász's striking proof of Kneser's conjecture from 1978 using the Borsuk-Ulam theorem provides a lower bound on the chromatic number χ(G) of a graph G. 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 Matoušek and Ziegler) to the Lovász complex. In the process, we analyze and clarify the combinatorics of the complexes involved and link their structure via several "intermediate" complexes.