Fourientation activities and the Tutte polynomial

TL;DR

This paper introduces a new framework for fourientations of graphs, leading to a 12-variable expansion of the Tutte polynomial that generalizes previous orientation activity expansions.

Contribution

It defines activities for fourientations that extend existing Tutte polynomial expansions, unifying and generalizing prior work.

Findings

Developed a 12-variable expansion of the Tutte polynomial

Unified previous orientation activity expansions by Las Vergnas and Gordon-Traldi

Provided a new perspective on graph orientations and subgraphs

Abstract

A fourientation of a graph $G$ is a choice for each edge of the graph whether to orient that edge in either direction, leave it unoriented, or biorient it. We may naturally view fourientations as a mixture of subgraphs and graph orientations where unoriented and bioriented edges play the role of absent and present subgraph edges, respectively. Building on work of Backman and Hopkins (2015), we show that given a linear order and a reference orientation of the edge set, one can define activities for fourientations of $G$ which allow for a new 12 variable expansion of the Tutte polynomial $T_G$. Our formula specializes to both the Las Vergnas (1984) orientation activites expansion of the Tutte polynomial and the generalized activities expansion of Gordon and Traldi (1990).