Edge colourings and topological graph polynomials

TL;DR

This paper introduces a new multivariate generating function for k-valuations in edge colourings, which forms a graph polynomial with recursive properties and unifies several existing graph polynomials, extending their applicability.

Contribution

It defines a new graph polynomial based on k-valuations that generalizes and unifies multiple well-known graph polynomials, with recursive and surface extension properties.

Findings

The generating function is a polynomial in k.

It satisfies a recursive deletion-contraction relation.

It extends the Penrose polynomial beyond plane graphs.

Abstract

A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollob\'as-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.