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.
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