A Hodge decomposition interpretation for the coefficients of the chromatic polynomial

Abstract

Let G G be a simple graph with n n nodes. The coloring complex of G G , as defined by Steingrimsson, has r r -faces consisting of all ordered set partitions, ( B 1 , … , B r + 2 ) (B_1, \ldots ,B_{r+2}) in which at least one B i B_i contains an edge of G G . Jonsson proved that the homology H ∗ ( G ) H_{*}(G) of the coloring complex is concentrated in the top degree. In addition, Jonsson showed that the dimension of the top homology is one less than the number of acyclic orientations of G G . In this paper, we show that the Eulerian idempotents give a decomposition of the top homology of G G into n − 1 n-1 components H n − 3 ( j ) ( G ) H_{n-3}^{(j)}(G) . We go on to prove that the dimensions of the Hodge pieces of the homology are equal to the absolute values of the coefficients of the chromatic polynomial of G G . Specifically, if we write χ G ( λ ) = ( ∑ j = 1 n − 1 c j ( − 1 ) n − j λ j ) + λ n \chi _G(\lambda ) = (\sum _{j=1}^{n-1} c_j (-1)^{n-j} \lambda ^j) + \lambda ^n , then d i m ( H n − 3 ( j ) ( G ) ) = c j dim(H_{n-3}^{(j)}(G)) = c_j .