Some results on total weight choosability

A graph [Formula: see text] is called [Formula: see text]-choosable if for any total list assignment [Formula: see text] which assigns to each vertex [Formula: see text] a set [Formula: see text] of [Formula: see text] real numbers, and assigns to each edge [Formula: see text] a set [Formula: see text] of [Formula: see text] real numbers, there is a mapping [Formula: see text] such that [Formula: see text] for any [Formula: see text] and for any two adjacent vertices [Formula: see text], [Formula: see text], where [Formula: see text] denotes the set of incident edges of a vertex [Formula: see text]. An [Formula: see text]-graph is a graph with [Formula: see text] vertices and [Formula: see text] edges. In this paper, we characterize sufficient conditions for [Formula: see text]-choosability and demonstrate that connected [Formula: see text]-graphs with [Formula: see text] or [Formula: see text] are both [Formula: see text]-choosable and [Formula: see text]-choosable. Furthermore, we establish [Formula: see text]-choosability for graphs obtained through specific graph operations. Our results contribute to the conjectures proposed by Wong and Zhu regarding total weight choosability.