Graph polynomials: some questions on the edge

TL;DR

This paper explores fundamental questions about graph polynomials, including their reduction relations, structural properties, and introduces new polynomials based on partial colorings, aiming to advance the theoretical understanding of graph invariants.

Contribution

It raises key open questions on the nature of graph polynomials, discusses their properties, and introduces new polynomials related to partial graph colorings.

Findings

Identification of conditions for reduction relations in graph polynomials

Analysis of how properties like factorisation relate to graph structure

Introduction of new polynomials based on partial colorings and their basic properties

Abstract

We raise some questions about graph polynomials, highlighting concepts and phenomena that may merit consideration in the development of a general theory. Our questions are mainly of three types: When do graph polynomials have reduction relations (simple linear recursions based on local operations), perhaps in a wider class of combinatorial objects? How many levels of reduction relations does a graph polynomial need in order to express it in terms of trivial base cases? For a graph polynomial, how are properties such as equivalence and factorisation reflected in the structure of a graph? We illustrate our discussion with a variety of graph polynomials and other invariants. This leads us to reflect on the historical origins of graph polynomials. We also introduce some new polynomials based on partial colourings of graphs and establish some of their basic properties.