Partial graph orientations and the Tutte polynomial

Abstract

Gessel and Sagan investigated the Tutte polynomial, TG(x,y) using depth-first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is 2m−n+1TG(3,1/2). We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is 2n−1TG(1/2,3). We then prove that the number of partial orientations modulo cycle reversals is 2gTG(3,1) and the number of partial orientations modulo cut reversals is 2n−1TG(1,3). To prove these results, we introduce cut and cycle-minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals, extending known representatives for full orientations introduced by Greene and Zaslavksy. We then introduce distinguished partial orientations representing a given indegree sequence. We utilize these partial orientations to derive the Ehrhart polynomial of the win vector polytope, and give a combinatorial interpretation of its volume, thus answering a question of Bartels, Mount, and Welsh. We conclude with edge chromatic generalizations of the quantities presented, which allow for a new interpretation of the reliability polynomial for all probabilities p with 0<p<1/2.