The Topology of the Coloring Complex

In a recent paper, E. Steingŕımsson associated to each simple graph G a simplicial complex ∆G denoted as the coloring complex of G. Certain nonfaces of ∆G correspond in a natural manner to proper colorings of G. Indeed, the h-vector is an affine transformation of the chromatic polynomial χG of G, and the reduced Euler characteristic is, up to sign, equal to |χG(−1)| − 1. We show that ∆G is constructible and hence Cohen-Macaulay. Moreover, we introduce two subcomplexes of the coloring complex, denoted as polar coloring complexes. The hvectors of these complexes are again affine transformations of χG, and their Euler characteristics coincide with χG(0) and −χG(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.