A note on entire choosability of plane graphs

A plane graph is called entirely kk-choosable if for any list assignment LL such that ∣L(x)∣=k∣L(x)∣=k for each x∈V(G)∪E(G)∪F(G)x∈V(G)∪E(G)∪F(G), we can assign each element xx a color from its list such that any two elements that are adjacent or incident receive distinct colors. Wang and Lih (2008) [5] conjectured that every plane graph is entirely (Δ+4)(Δ+4)-choosable and showed that the conjecture is true if Δ≥12Δ≥12. In this note, we prove that (1)(1) Every plane graph GG with Δ≥7Δ≥7 is entirely (Δ+4)(Δ+4)-choosable. (2)(2) Every plane graph GG with Δ≥6Δ≥6 is entirely (Δ+5)(Δ+5)-choosable.