Total Weight Choosability of Trees

A total-weighting of a graph G = (V,E) is a mapping f which assigns to each element y ∈ V ∪ E a real number f(y) as the weight of y. A total-weighting f of G is proper if the colouring φf of the vertices of G defined as φf (v) = f(v) + ∑ e∋v f(e) is a proper colouring of G, i.e., φf (v) ̸= φf (u) for any edge uv. For positive integers k and k′, a graph G is called (k, k′)-total-weight-choosable if whenever each vertex v is given k permissible weights and each edge e is given k′ permissible weights, there is a proper total-weighting f of G which uses only permissible weights on each element y ∈ V ∪E. It is known that every tree is (2, 2)-total-weight-choosable and every tree other than K2 is (1, 3)-total-weight-choosable. However, the problem of determining which trees are (1, 2)-total-weight-choosable remained open. In this talk, I present the result in a joint paper with Gerard Jennhwa Chang, Guan-Huei Duh and Tsai-Lien Wong, in which we solve this problem and characterizes all (1, 2)-total-weight-choosable trees. Based on this characterization, we give an algorithm that determines in linear time whether a given tree is (1, 2)-total-weight-choosable.