Partial Graph Orientations and the Tutte Polynomial

TL;DR

This paper explores the enumeration of various classes of partial graph orientations related to the Tutte polynomial, providing new proofs, classifications, and connections to algebraic and probabilistic graph invariants.

Contribution

It introduces a deletion-contraction proof for counting acyclic and strongly connected partial orientations and defines minimal representatives for orientations modulo cycle and cut reversals.

Findings

Number of acyclic partial orientations: $2^gT(3,1/2)$

Number of strongly connected partial orientations: $2^{n-1}T(1/2,3)$

Edge chromatic generalizations relate to the reliability polynomial

Abstract

Gessel and Sagan investigated the Tutte polynomial, $T(x,y)$ using depth first search, and applied their techniques to show that the number of acyclic partial orientations of a graph is $2^gT(3,1/2)$. We provide a short deletion-contraction proof of this result and demonstrate that dually, the number of strongly connected partial orientations is $2^{n-1}T(1/2,3)$. We then prove that the number of partial orientations modulo cycle reversals is $2^gT(3,1)$ and the number of partial orientations modulo cut reversals is $2^{n-1}T(1,3)$. To prove these results, we introduce cut and cycle minimal partial orientations which provide distinguished representatives for partial orientations modulo cut and cycle reversals. These extend classes of total orientations introduced by Gioan, and Greene and Zaslavksy, and we highlight a close connection with graphic and cographic Lawrence ideals. 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 $0 < p <1/2$.