List Coloring in the Absence of Two Subgraphs

A list assignment of a graph G = (V,E) is a function 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 L, i.e., whether G has a mapping c : V → {1, 2, . . .} such that (i) c(u) 6= 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 {H1, H2}, then G is called (H1, H2)-free. We completely characterize the complexity of List Coloring for (H1, H2)-free graphs.