|Authors:||Živaljević, Rade||Affiliations:||Mathematical Institute of the Serbian Academy of Sciences and Arts||Title:||WI-posets, graph complexes and ℤ2-equivalences||Journal:||Journal of Combinatorial Theory. Series A||Volume:||111||Issue:||2||First page:||204||Last page:||223||Issue Date:||1-Jan-2005||Rank:||M22||ISSN:||0097-3165||DOI:||10.1016/j.jcta.2004.12.002||Abstract:||
An evergreen theme in topological graph theory is the study of graph complexes, (Proof of the Lovász conjecture, arXiv:math.CO/ 0402395, 2, 2004; J. Combin. Theory Ser. A 25 (1978) 319-324; Using the Borsuk-Ulam Theorem, Lectures on Topological Methods in Combinatorics and Geometry, Springer Universitext, Berlin, 2003; ). Many of these complexes are ℤ 2 -spaces and the associated ℤ 2 -index Ind ℤ2 (X) is an invariant of great importance for estimating the chromatic numbers of graphs. We introduce WI-posets (Definition 2) as intermediate objects and emphasize the importance of Bredon's theorem (Theorem 9) which allows us to use standard tools of topological combinatorics for comparison of ℤ 2 -homotopy types of ℤ 2 -posets. Among the consequences of general results are known and new results about ℤ 2 -homotopy types of graph complexes. It turns out that, in spite of great variety of approaches and definitions, all ℤ 2 -graph complexes associated to G can be viewed as avatars of the same object, as long as their ℤ 2 -homotopy types are concerned. Among the applications are a proof that each finite, free ℤ 2 -complex is a graph complex and an evaluation of ℤ 2 -homotopy types of complexes Ind (C n ) of independence sets in a cycle C n .
|Keywords:||Bredon's theorem | Graph complexes | WI-posets||Publisher:||Elsevier|
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