Total weight choosability for Halin graphs

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, Σe ∈ E(v) φ(e)+φ(v) ≠ Σe ∈ 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. As a strenghtening of the well-known 1-2-3 conjecture, it was conjectured in [Wong and Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198-212] that every graph without isolated edge is (1,3)-choosable. It is easy to verified this conjecture for trees, however, to prove it for wheels seemed to be quite non-trivial. In this paper, we develop some tools and techniques which enable us to prove this conjecture for generalized Halin graphs.