Planar graphs without normally adjacent short cycles

Abstract

Let $\mathcal{G}$ be the class of plane graphs without triangles normally adjacent to $8^{-}$-cycles, without $4$-cycles normally adjacent to $6^{-}$-cycles, and without normally adjacent $5$-cycles. In this paper, it is shown that every graph in $\mathcal{G}$ is $3$-choosable. Instead of proving this result, we directly prove a stronger result in the form of ``weakly'' DP-$3$-coloring. The main theorem improves the results in [J. Combin. Theory Ser. B 129 (2018) 38--54; European J. Combin. 82 (2019) 102995]. Consequently, every planar graph without $4$-, $6$-, $8$-cycles is $3$-choosable, and every planar graph without $4$-, $5$-, $7$-, $8$-cycles is $3$-choosable. In the third section, using almost the same technique, we prove that the vertex set of every graph in $\mathcal{G}$ can be partitioned into an independent set and a set that induces a forest, which strengthens the result in [Discrete Appl. Math. 284 (2020) 626--630]. In the final section, tightness is discussed.

17 pages, 3 figures

References in corpus (4)

• Cover and variable degeneracy
• DP-4-coloring of planar graphs with some restrictions on cycles
• Variable degeneracy on toroidal graphs
• An extension of Thomassen's result on choosability

Cited by in corpus (3)

• Weak degeneracy of planar graphs without 4- and 6-cycles
• DP-4-coloring of planar graphs with some restrictions on cycles
• DP-3-coloring of planar graphs without certain cycles