Total weight choosability of graphs with bounded maximum average degree

A total weighting of a graph G is a function ϕ that assigns a weight to each vertex and each edge of G. The vertex-sum of a vertex v with respect to ϕ is Sϕ(v)=ϕ(v)+∑e∈E(v)ϕ(e), where E(v) is the set of edges incident to v. A total weighting is proper if adjacent vertices have distinct vertex-sums. A graph G is (k,k′)-choosable if the following is true: Whenever each vertex x is assigned a set L(x) of k real numbers and each edge e is assigned a set L(e) of k′ real numbers, there is a proper total weighting ϕ of G with ϕ(y)∈L(y) for all y∈V(G)∪E(G). In this paper, we prove that for p∈{5,7,11}, a graph G without isolated edges and with mad(G)≤p−1 is (1,p)-choosable. In particular, triangle-free planar graphs are (1,5)-choosable.