List backbone colouring of graphs

Suppose that G is a graph and that H is a subgraph of G. Let L be a mapping that assigns to each vertex v of G a set L(v) of positive integers. We say that (G,H) is backboneL-colourable if there is a proper vertex colouring c of G such that c(v)@?L(v) for all [email protected]?V, and |c(u)-c(v)|>=2 for every edge uv in H. We say that (G,H) is backbone k-choosable if (G,H) is backbone L-colourable for any list assignment L with |L(v)|=k for all [email protected]?V(G). The backbone choice number of (G,H), denoted by ch"B"B(G,H), is the minimum k such that (G,H) is backbone k-choosable. The concept of a backbone choice number is a generalization of both the choice number and the L(2,1)-choice number. Precisely, if E(H)[email protected]?, then ch"B"B(G,H)=ch(G), where ch(G) is the choice number of G; if G=H^2, then ch"B"B(G,H) is the same as the L(2,1)-choice number of H. In this article, we first show that, if |L(v)|=d"G(v)+2d"H(v), then (G,H) is L-colourable, unless E(H)[email protected]? and each block of G is a complete graph or an odd cycle. This generalizes a result of Erdos, Rubin, and Taylor on degree-choosable graphs. Second, we prove that ch"B"B(G,H)=