Total weight choosability for Halin graphs

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 , 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.