The topology of the coloring complex

Abstract

In a recent paper, E. Steingrímsson associated to each simple graph G a simplicial complex Δ G, referred to as the coloring complex of G. Certain nonfaces of Δ G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial χ G of G, and the reduced Euler characteristic is, up to sign, equal to |χG (-1)|-1. We show that Δ G is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, referred to as polar coloring complexes. The h-vectors of these complexes are again affine transformations of χG, and their Euler characteristics coincide with χ′G (0) and -χ′G (1), respectively. We show for a large class of graphs-including all connected graphs-that polar coloring complexes are constructible. Finally, the coloring complex and its polar subcomplexes being Cohen-Macaulay allows for topological interpretations of certain positivity results about the chromatic polynomial due to N. Linial and I. M. Gessel. © 2005 Springer Science + Business Media, Inc.