The homology of the cyclic coloring complex of simple graphs

Let G be a simple graph on n vertices, and let χG(λ) denote the chromatic polynomial of G. In this paper, we define the cyclic coloring complex, ∆(G), and determine the dimensions of its homology groups for simple graphs. In particular, we show that if G has r connected components, the dimension of (n− 3)rd homology group of ∆(G) is equal to (n− (r+ 1)) plus 1 r! |χG(0)|, where χG is the rth derivative of χG(λ). We also define a complex ∆(G)C , whose r-faces consist of all ordered set partitions [B1, ..., Br+2] where none of the Bi contain an edge of G and where 1 ∈ Bi. We compute the dimensions of the homology groups of this complex, and as a result, obtain the dimensions of the multilinear parts of the cyclic homology groups of C[x1, ..., xn]/{xixj |ij is an edge of G}. We show that when G is a connected graph, the homology of ∆(G)C has nonzero homology only in dimension n − 2, and the dimension of this homology group is |χG(0)|. In this case, we provide a bijection between a set of homology representatives of ∆(G)C and the acyclic orientations of G with a unique source at v, a vertex of G.