List backbone colouring of graphs

Suppose G is a graph and 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 (G,H) is backbone L-colourable if there is a proper vertex colouring c of G such that c(v) ∈ L(v) for all v ∈ V , and |c(u) − c(v)| > 2 for every edge uv in H . We say (G,H) is backbone k-choosable if (G,H) is backbone Lcolourable for any list assignment L with |L(v)| = k for all v ∈ V (G). The backbone choice number of (G,H), denoted by chBB(G,H), is the minimum k such that (G,H) is backbone k-choosable. The concept of backbone choice number is a generalization of both the choice number, and the L(2, 1)-choice number. Precisely, if E(H) = ∅ then chBB(G,H) = ch(G), where ch(G) is the choice number of G; if G = H then chBB(G,H) is the same as the L(2, 1)choice number of H . In this article, we first show that if |L(v)| = dG(v) + 2dH(v) then (G,H) is L-colourable, unless E(H) = ∅ and each block of G is a complete graph or an odd cycle. This generalizes a result of Erdős, Rubin and Taylor on degree-choosable graphs. Secondly, we prove that chBB(G,H) 6 max {bmad(G)c+ 1, bmad(G) + 2mad(H)c}, where mad(G) is the maximum average degree of a graph G. Finally, we establish various upper bounds on chBB(G,H) in terms of ch(G). In particular, we prove that for a k-choosable graph G, chBB(G,H)6 3k if every component of H is unicyclic; chBB(G,H) 6 2k if H is a matching; and chBB(G,H) 6 2k + 1 if H is a disjoint union of paths with lengths at most two.