Graphical Eulerian numbers and chromatic generating functions

Abstract

In this paper some further properties of the coefficients of a chromatic generating function introduced by Linial are proved. A combinatorial interpretation of these numbers is given by specializing some results of Stanley on posets to surjective n-colorings of a graph G of order n compatible with linear orders on V(G) which extend acyclic orientations of G. If G is composed from n isolated vertices these coefficients are classical Eulerian numbers A(n, k). © 1987.