Every graph is (2,3)-choosable.

A total weighting of a graph G is a mapping ϕ that assigns to each element z ∈ V ( G)∪ E( G) a weight ϕ( z). A total weighting ϕ is proper if for any two adjacent vertices u and v, ∑ ϕ( e)+ ϕ( u)≠∑ ϕ( e)+ ϕ( v). This paper proves that if each edge e is given a set L( e) of 3 permissible weights, and each vertex v is given a set L( v) of 2 permissible weights, then G has a proper total weighting ϕ with ϕ( z) ∈ L( z) for each element z ∈ V ( G)∪ E( G). [ABSTRACT FROM AUTHOR]