Total list weighting of graphs with bounded maximum average degree

Abstract

A proper total weighting of a graph G is a mapping ϕ which assigns to each vertex and each edge of G a real number as its weight so that for any edge uv of G, ∑e∈E(v)ϕ(e)+ϕ(v)≠∑e∈E(u)ϕ(e)+ϕ(u). A (k,k′)-list assignment of G is a mapping L which assigns to each vertex v a set L(v) of k permissible weights and to each edge e a set L(e) of k′ permissible weights. An L-total weighting is a total weighting ϕ with ϕ(z)∈L(z) for each z∈V(G)∪E(G). A graph G is called (k,k′)-choosable if for every (k,k′)-list assignment L of G, there exists a proper L-total weighting. It was proved in Tang and Zhu (2017) that if p∈{5,7,11}, a graph G without isolated edges and with mad(G)≤p−1 is (1,p)-choosable. In this paper, we strengthen this result by showing that for any prime p, a graph G without isolated edges and with mad(G)≤p−1 is (1,p)-choosable.