Abstract
We extend the concept of the Alon–Tarsi number for unsigned graph to signed one. Moreover, we introduce the modulo Alon–Tarsi number for a prime number p. We show that both the Alon–Tarsi number and modulo Alon–Tarsi number of a signed planar graph \((G,\sigma )\) are at most 5, where the former result generalizes Zhu’s result for unsigned case and the latter one implies that \((G,\sigma )\) is \({\mathbb {Z}}_5\)-colorable.
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We would like to thank the anonymous referee for many helpful comments.
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Supported by the National Natural Science Foundation of China under Grant nos. 11471273 and 11561058.
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Wang, W., Qian, J. & Abe, T. Alon–Tarsi Number and Modulo Alon–Tarsi Number of Signed Graphs. Graphs and Combinatorics 35, 1051–1064 (2019). https://doi.org/10.1007/s00373-019-02056-9
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DOI: https://doi.org/10.1007/s00373-019-02056-9
Keywords
- Signed graph
- Group coloring
- \({\mathbb {Z}}_{p}\)-coloring
- Planar graph
- List coloring
- Combinatorial Nullstellensatz
- Alon–Tarsi number