Planar graphs without normally adjacent short cycles

Abstract

Let 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 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 Dvořák and Postle [J. Comb. Theory, Ser. B 129 (2018) 38–54] [5]; Liu and Li [Eur. J. Comb. 82 (2019) 102995] [13]. 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 G can be partitioned into an independent set and a set that induces a forest, which strengthens the result in Liu and Yu [Discrete Appl. Math. 284 (2020) 626–630] [17]. In the final section, tightness is discussed.