On sufficient conditions for planar graphs to be 5-flexible

Abstract

In this paper, we study the flexibility of two planar graph classes H 1 , H 2 , where H 1 , H 2 denote the set of all hopper-free planar graphs and house-free planar graphs, respectively. Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G ∈ H 1 or G ∈ H 2 such that all lists have size at least 5, then there exists an L-coloring respecting at least a constant fraction of the preferences.