On Sufficient Conditions for Planar Graphs to be 5-Flexible

• DOI: 10.1007/s00373-022-02480-4
• Publication Date: 2022-03-14T16:05:32Z
• Authors: Fan Yang
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO)

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.