(1,0,0)-colorability of planar graphs without cycles of length 4 or 6

Jin, Ligang; Kang, Yingli; Li, Peipei; Wang, Yingqian

doi:10.1016/j.disc.2021.112758

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2022

2023

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“…On the other hand, in 2022 Kang et al [14] showed that every planar graph without 4-cycles and 6-cycles has an (F 1 , I, I)-partition (equivalently, a (1, 0, 0)-coloring). Inspired by two above results, we put forth the following conjecture.…”

Section: Introductionmentioning

confidence: 99%

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

Sittitrai¹,

Nakprasit²

2022

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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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Section: Introductionmentioning

confidence: 99%