Abstract
Let G be a graph. For two positive integers d and h, a (d, h)-decomposition of G is a pair (G, H) such that H is a subgraph of G of maximum degree at most h and D is an acyclic orientation of \(G-E(H)\) of maximum out-degree at most d. A graph G is (d, h)-decomposable if G has a (d, h)-decomposition. In this paper, we prove that every planar graph without intersecting 3-cycles and adjacent \(4^-\)-cycles is (2, 1)-decomposable. As a corollary, we obtain that every planar graph without intersecting 3-cycles and adjacent \(4^-\)-cycles has a matching M such that \(G-M\) is 2-degenerate and hence \(G-M\) is DP-3-colorable and Alon-Tarsi number of \(G-M\) is at most 3.
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Author Xiangwen Li was supported by the NSFC (12031018).
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Tian, F., Li, X. A (2, 1)-Decomposition of Planar Graphs Without Intersecting 3-Cycles and Adjacent \(4^-\)-Cycles. Graphs and Combinatorics 39, 115 (2023). https://doi.org/10.1007/s00373-023-02708-x
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DOI: https://doi.org/10.1007/s00373-023-02708-x