The degeneracy and Alon-Tarsi number under $F$-sum operations
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arXiv
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| Main Authors: | , , |
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| Format: | Preprint |
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2026
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| _version_ | 1866910044118843392 |
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| author | Li, Zhiguo Jiao, Zhentao Shao, Zeling |
| author_facet | Li, Zhiguo Jiao, Zhentao Shao, Zeling |
| contents | The Alon-Tarsi number of a graph $ G $ is the smallest $ k $ such that there exists an orientation $ D $ of $ G $ with maximum outdegree $ k - 1 $ satisfying that the number of even Eulerian subgraphs is different from the number of odd Eulerian subgraphs. The degeneracy of a graph $ G $ is the maximum value of the minimum degree over all subgraphs of $ G $. In this paper, we obtain a characterization of graphs with $AT(G)=2$ for any graph $G$, and study the Alon-Tarsi number of $F$-sum in terms of degeneracy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_06747 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The degeneracy and Alon-Tarsi number under $F$-sum operations Li, Zhiguo Jiao, Zhentao Shao, Zeling Combinatorics 05C15 The Alon-Tarsi number of a graph $ G $ is the smallest $ k $ such that there exists an orientation $ D $ of $ G $ with maximum outdegree $ k - 1 $ satisfying that the number of even Eulerian subgraphs is different from the number of odd Eulerian subgraphs. The degeneracy of a graph $ G $ is the maximum value of the minimum degree over all subgraphs of $ G $. In this paper, we obtain a characterization of graphs with $AT(G)=2$ for any graph $G$, and study the Alon-Tarsi number of $F$-sum in terms of degeneracy. |
| title | The degeneracy and Alon-Tarsi number under $F$-sum operations |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2603.06747 |