Edge-Face List Coloring of Halin Graphs

A plane graph G is k-edge-face colorable if the elements of \(E(G)\cup F(G)\) can be colored with k colors such that any two adjacent or incident elements receive different colors. G is edge-face L-list colorable if for a given list assignment \(L=\{L(x){\mid }x\in E(G)\cup F(G)\}\), there exists a proper edge-face coloring \(\pi \) of G such that \(\pi (x)\in L(x)\) for all \(x\in E(G)\,\cup \,F(G)\). If G is edge-face L-list colorable for any list assignment with \(|L(x)|=k\) for all \(x\in E(G)\,\cup \,F(G)\), then G is edge-face k-choosable. The edge-face list chromatic number is defined to be the smallest integer k such that G admits an edge-face k-list coloring.