Distinctive power and comparability of Harary polynomial

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 $\chi_{\mathcal{P}}$.