Counting List Colorings of Unlabeled Graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917242622443520 |
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| author | Kaul, Hemanshu Mudrock, Jeffrey A. |
| author_facet | Kaul, Hemanshu Mudrock, Jeffrey A. |
| contents | The classic enumerative functions for counting colorings of a graph $G$, such as the chromatic polynomial $P(G,k)$, do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph $\mathcal{G}$, $P(\mathcal{G}, k)$. Determining $P(\mathcal{G}, k)$ amounts to counting colorings under the action of automorphisms of $\mathcal{G}$. In this paper, we consider the problem of counting list colorings of unlabeled graphs. We extend Hanlon's definition to the list context and define the unlabeled list color function, $P_\ell(\mathcal{G}, k)$, of an unlabeled graph $\mathcal{G}$. In this context, we pursue a fundamental question whose analogues have driven much of the research on counting list colorings and its generalizations: For a given unlabeled graph $\mathcal{G}$, does $P_\ell(\mathcal{G}, k) = P(\mathcal{G}, k)$ when $k$ is large enough? We show the answer to this question is yes for a large class of unlabeled graphs that include point-determining graphs (also known as twin-free graphs, irreducible graphs, and mating graphs). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_06063 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Counting List Colorings of Unlabeled Graphs Kaul, Hemanshu Mudrock, Jeffrey A. Combinatorics 20B25, 05C15, 05C30, 05A99 The classic enumerative functions for counting colorings of a graph $G$, such as the chromatic polynomial $P(G,k)$, do so under the assumption that the given graph is labeled. In 1985, Hanlon defined and studied the chromatic polynomial for an unlabeled graph $\mathcal{G}$, $P(\mathcal{G}, k)$. Determining $P(\mathcal{G}, k)$ amounts to counting colorings under the action of automorphisms of $\mathcal{G}$. In this paper, we consider the problem of counting list colorings of unlabeled graphs. We extend Hanlon's definition to the list context and define the unlabeled list color function, $P_\ell(\mathcal{G}, k)$, of an unlabeled graph $\mathcal{G}$. In this context, we pursue a fundamental question whose analogues have driven much of the research on counting list colorings and its generalizations: For a given unlabeled graph $\mathcal{G}$, does $P_\ell(\mathcal{G}, k) = P(\mathcal{G}, k)$ when $k$ is large enough? We show the answer to this question is yes for a large class of unlabeled graphs that include point-determining graphs (also known as twin-free graphs, irreducible graphs, and mating graphs). |
| title | Counting List Colorings of Unlabeled Graphs |
| topic | Combinatorics 20B25, 05C15, 05C30, 05A99 |
| url | https://arxiv.org/abs/2409.06063 |