The Dichromate and Orientations of a Graph

Abstract

Internal and external activities are defined for any orientation of a graph relative to a fixed labelling of its edges. It is shown that the number of such orientations of having internal activity r and external activity s is 2r+sXrs where Xrs is the coefficient of xrys in the dichromate It follows that the number of orientations of in which the resulting digraph is acyclic is given by , where is the chromatic polynomial associated with .