Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths

Pongpat Sittitrai; Kittikorn Nakprasit

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arXiv · 2203.06466

Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths

Pongpat Sittitrai·Kittikorn Nakprasit

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Abstract

In this paper, we show that every planar graph without $4$-cycles and $6$-cycles has a partition of its vertex set into two sets, where one set induces a forest, and the other induces a forest with maximum degree at most $2$ (equivalently, a disjoint union of paths). Note that we can partition the vertex set of a forest into two independent sets. However, a pair of independent sets combined may not induce a forest. Thus our result extends the result of Wang and Xu (2013) stating that the vertex set of every planar graph without $4$-cycles and $6$-cycles can be partitioned into three sets, where one induces a graph with maximum degree two, and the remaining two are independent sets.

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Pongpat Sittitrai, Kittikorn Nakprasit. 2022-03-12. Partitioning planar graphs without 4-cycles and 6-cycles into a forest and a disjoint union of paths. https://arxiv.org/abs/2203.06466

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