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| Natura: | Preprint |
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2025
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| Accesso online: | https://arxiv.org/abs/2512.22556 |
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| _version_ | 1866912791164616704 |
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| author | Makowsky, Johann A. |
| author_facet | Makowsky, Johann A. |
| contents | 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 $χ_{\mathcal{P}}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22556 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distinctive power and comparability of Harary polynomial Makowsky, Johann A. Combinatorics 05C15, 05C31, 03C35 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 $χ_{\mathcal{P}}$. |
| title | Distinctive power and comparability of Harary polynomial |
| topic | Combinatorics 05C15, 05C31, 03C35 |
| url | https://arxiv.org/abs/2512.22556 |