List Total Weighting of Graphs

Abstract

A graph G=(V, E) is (k, k′)-total weight choosable if the following is true: For any (k, k′)-total list assignment L that 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, there is a proper L-total weighting, i.e., 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, ∑ _e∈E(u) f(e)+f(u)≠∑ _e∈E(v) f(e)+f(v). This Paper introduces a method, the max-min weighting method, for finding proper L-total weightings of graphs. Using this method, we prove that complete multipartite graphs of the form K _{n,m,1,1,...,1} are (2,2)-total weight choosable and complete bipartite graphs other than K ₂ are (1,2)-total weight choosable.