### 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.

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PDFDOI: http://dx.doi.org/10.5614/ejgta.2021.9.1.2

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