Distinctive power and comparability of Harary polynomial

TL;DR

This work develops a cohesive framework for Harary polynomials $\chi_{\mathcal{P}}(G;k)$, examining their distinguishing power relative to other graph polynomials and illustrating when graphs admit $\mathcal{P}$-mates. By formalizing $F$-mates, $d.p.$-equivalence, and concepts like Compton-Gessel classes and heredity, the paper derives broad results on the existence and abundance of mates, the (non)completeness of Harary polynomials, and the comparability of different $\chi_{\mathcal{P}}$-families. It also explores generating-function viewpoints and random-graph behavior to identify weakly distinguishing Harary polynomials, including the $F_r$ family, and presents concrete theorems on delta-type results and the prevalence of mates in various regimes. Overall, the paper clarifies the limitations and potential of Harary polynomials for graph discrimination, guiding future work on their logical definability, DP-incomparability, and asymptotic distinguishability in probabilistic graph models.

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 $χ_{\mathcal{P}}(G; k)$ the number of $\mathcal{P}$-colorings of $G$ with at most $k$ colors. $χ_{\mathcal{P}}(G; k)$ 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 $χ_{\mathcal{P}}(G; k)$ definable in Monadic Second Order Logic and under which conditions is $χ_{\mathcal{P}}(G; k)$ a chromatic invariant. Let $\mathcal{P}, \mathcal{Q}$ be two graph properties. Two graphs $G, H$ are $\mathcal{P}$-mates if $χ_{\mathcal{P}}(G; k) = χ_{\mathcal{P}}(H; k)$. $χ_{\mathcal{Q}}$ is at least as distinctive as $χ_{\mathcal{P}}$, $χ_{\mathcal{P}} \leq χ_{\mathcal{Q}}$, if for all graphs $G, H$ we have that $χ_{\mathcal{Q}}(G; k) = χ_{\mathcal{Q}}(H; k)$ implies $χ_{\mathcal{P}}(G; k) = χ_{\mathcal{P}}(H; k)$. 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 $χ_{\mathcal{Q}}$ is at least as distinctive as $χ_{\mathcal{P}}$.

Theorems & Definitions (83)

• Definition 1.1
• Theorem 1.2
• Definition 1.3: Distinctive power
• Proposition 1.4
• proof
• Proposition 1.5
• proof
• Lemma 2.2
• Proposition 2.3
• Theorem 2.4