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 , Σ e ∈ E ( v ) φ( e )+φ( v ) ≠ Σ 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. 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.