On Sufficient Conditions for Planar Graphs to be 5-Flexible

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 is an element of H-1 or G is an element of 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.