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.
Categories: math.CO