On sufficient conditions for planar graphs to be 5-flexible

TL;DR

This paper establishes sufficient conditions under which certain classes of planar graphs, specifically hopper-free and house-free, are 5-flexible in list coloring, ensuring many preferences are satisfied.

Contribution

It proves that hopper-free and house-free planar graphs with list sizes of at least 5 are 5-flexible, extending understanding of graph coloring flexibility.

Findings

Hopper-free planar graphs are 5-flexible.

House-free planar graphs are 5-flexible.

A constant fraction of preferences can be respected in these classes.

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.