On sufficient conditions for planar graphs to be 5-flexible

Abstract

In this paper, we study the flexibility of two planar graph classes ${\mathcal{H}}_1$, ${\mathcal{H}}_2$, where ${\mathcal{H}}_1$, ${\mathcal{H}}_2$ 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 $G\in {\mathcal{H}}_1$ or $G\in {\mathcal{H}}_2$ such that all lists have size at least $5$, then there exists an $L$-coloring respecting at least a constant fraction of the preferences.