On total weight choosability of graphs

Abstract

For a graph G with vertex set V and edge set E, a (k,k′)-total list assignment L of G assigns to each vertex v a set L(v) of k real numbers as permissible weights, and assigns to each edge e a set L(e) of k′ real numbers as permissible weights. If for any (k,k′)-total list assignment L of G, there exists a mapping f:V∪E→ℝ such that f(y)∈L(y) for each y∈V∪E, and for any two adjacent vertices u and v, ∑y∈N(u)f(uy)+f(u)≠∑x∈N(v)f(vx)+f(v), then G is (k,k′)-total weight choosable. It is conjectured by Wong and Zhu that every graph is (2,2)-total weight choosable, and every graph with no isolated edges is (1,3)-total weight choosable.