Abstract
Let \(c_1,\ldots , c_k\) be k non-negative integers. A graph G is \((c_1, \ldots , c_k)\)-colorable if the vertex set of G can be partitioned into k sets \(V_1, \ldots , V_k\), such that the induced subgraph \(G[V_i]\) has maximum degree at most \(c_i\) for \(i\in [k]\). Denote by \(\mathscr {F}\) the family of planar graphs without triangles adjacent to cycles of length 5 and 7. This paper proves that if \(G\in \mathscr {F}\), then G is (1, 1, 1)-colorable. As a corollary, every graph in \(\mathscr {F}\) can be decomposed into a matching and a 3-colorable graph.
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The authors would like to thank two anonymous referees for their careful reading and suggestions which greatly improved the presentation of the paper.
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This work is supported by National Natural Science Foundation of China (Nos. 12261094, 11501256 and 11701496).
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Huang, Z., Yang, F. & Zhang, X. Decomposing planar graphs without triangular short cycles into a matching and a 3-colorable graph. J. Appl. Math. Comput. 70, 1723–1746 (2024). https://doi.org/10.1007/s12190-024-02028-0
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DOI: https://doi.org/10.1007/s12190-024-02028-0