List Coloring in the Absence of Two Subgraphs

A list assignment of a graph G = (V,E) is a function \({\cal L}\) that assigns a list L(u) of so-called admissible colors to each u ∈ V. The List Coloring problem is that of testing whether a given graph G = (V,E) has a coloring c that respects a given list assignment \({\cal L}\), i.e., whether G has a mapping c: V → {1,2,…} such that (i) c(u) ≠ c(v) whenever uv ∈ E and (ii) c(u) ∈ L(u) for all u ∈ V. If a graph G has no induced subgraph isomorphic to some graph of a pair {H 1,H 2}, then G is called (H 1,H 2)-free. We completely characterize the complexity of List Coloring for (H 1,H 2)-free graphs.