Published online by Cambridge University Press: 20 November 2018
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
.