Abstract
Let \(G=(V(G), E(G))\) be a graph and \({\mathcal {G}}_i\) be a class of graphs for each \(i\in [k]\). A \(({\mathcal {G}}_1,\ldots , {\mathcal {G}}_k)\)-partition of G is a partition of V(G) into k sets \(V_1,\ldots , V_k\) such that, for each \(j\in [k]\), the graph \(G[V_j]\) induced by \(V_j\) is a graph in \({\mathcal {G}}_j\). In this paper, we prove that every planar graph without 4-cycles and 6-cycles admits an \(({\mathcal {F}}_2, {\mathcal {F}})\)-partition. As a corollary, V(G) can be partitioned into two sets \(V_1\) and \(V_2\) such that \(V_1\) induces a linear forest and \(V_2\) induces a forest if G is a planar graph without 4-cycles and 6-cycles.
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Acknowledgements
The authors would like to thank the anonymous reviewers for their comments which improved the original manuscript. This work is supported by the Science and Technology Project of Guangxi (Guike AD21220114), National Natural Science Foundation of China (Nos. 12261094, 12161010 and 11861069), and Youth Science Foundation of Guangxi (No.2019JJB110007).
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Huang, X., Huang, Z. & Lv, JB. Partitioning Planar Graphs without 4-Cycles and 6-Cycles into a Linear Forest and a Forest. Graphs and Combinatorics 39, 10 (2023). https://doi.org/10.1007/s00373-022-02605-9
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DOI: https://doi.org/10.1007/s00373-022-02605-9