A general notion of activity for the Tutte polynomial

TL;DR

This paper introduces a unified framework called Δ-activity that encompasses various notions of activity used to define the Tutte polynomial, clarifying their interrelations.

Contribution

It develops a general theory of activity for the Tutte polynomial, unifying previously distinct notions into a single coherent framework.

Findings

Unified the different notions of activity under the Δ-activity framework

Provided insights into the connections between various Tutte polynomial expressions

Enhanced understanding of the combinatorial interpretations of the Tutte polynomial

Abstract

In the literature can be found several descriptions of the Tutte polynomial of graphs. Tutte defined it thanks to a notion of activity based on an ordering of the edges. Thereafter, Bernardi gave a non-equivalent notion of the activity where the graph is embedded in a surface. In this paper, we see that other notions of activity can thus be imagined and they can all be embodied in a same notion, the $\Delta$-activity. We develop a short theory which sheds light on the connections between the different expressions of the Tutte polynomial.