Distinctive power and comparability of Harary polynomial

TL;DR

This paper investigates the properties and distinctions of Harary polynomials, which generalize chromatic polynomials based on graph colorings constrained by a property, focusing on their definability, invariance, and the concept of graph 'mates' with similar polynomial characteristics.

Contribution

It analyzes conditions under which Harary polynomials are definable, invariant, and how they compare in distinctiveness, extending previous work on their logical and algebraic properties.

Findings

Conditions for Harary polynomial definability in Monadic Second Order Logic

Criteria for invariance of Harary polynomials as chromatic invariants

Relationships between different Harary polynomials in terms of distinctiveness

Abstract

Let $\mathcal{P}$ be a graph property. A $\mathcal{P}$-coloring with at most $k$ colors is a coloring of the vertices of a simple graph $G$ such that each color class induces a graph in $\mathcal{P}$. Harary polynomials are generalizations of the chromatic polynomial for simple graphs based on conditional colorings. We denote by $\chi_{\mathcal{P}}(G; k)$ the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. $\chi_{\mathcal{P}}(G; k)$ is a polynomial in $\Z[k]$. A first paper studying Harary polynomials systematically was published in 2021 by O.Herscovici, J.A. Makowsky and V. Rakita. It studies under which conditions on $\mathcal{P}$ is $\chi_{\mathcal{P}}(G; k)$ definable in Monadic Second Order Logic and under which conditions is $\chi_{\mathcal{P}}(G; k)$ a chromatic invariant. Let $\mathcal{P}, \mathcal{Q}$ be two graph properties. Two graphs $G, H$ are $\mathcal{P}$-mates if $\chi_{\mathcal{P}}(G; k) = \chi_{\mathcal{P}}(H; k)$. $\chi_{\mathcal{Q}}$ is at least as distinctive as $\chi_{\mathcal{P}}$, $\chi_{\mathcal{P}} \leq \chi_{\mathcal{Q}}$, if for all graphs $G, H$ we have that $\chi_{\mathcal{Q}}(G; k) = \chi_{\mathcal{Q}}(H; k)$ implies $\chi_{\mathcal{P}}(G; k) = \chi_{\mathcal{P}}(H; k)$. In this paper we study under which conditions on $\mathcal{P}$ are there any (many) $\mathcal{P}$-mates and under which conditions on $\mathcal{P}, \mathcal{Q}$ is $\chi_{\mathcal{Q}}$ is at least as distinctive as $\chi_{\mathcal{P}}$.