Choosability with separation of planar graphs without prescribed cycles

Abstract

In terms of constraining the list assignment, one refinement of k-choosability is considered as choosability with separation. We call a graph (k, d)-choosable if it is colorable from lists of size k where adjacent vertices have at most d common colors in their lists. If two cycles have exactly one common edge, then they are said to be normally adjacent. In this article, it is shown that planar graphs without 5-cycles and normally adjacent 4-cycles are (3,1)-choosable. This extends a result that planar graphs without 5- and 6-cycles are (3,1)-choosable (Choi et al. (2016))