List total weighting of graphs

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) 6= ∑ 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 Kn,m,1,1,...,1 are (2, 2)-total weight choosable and complete bipartite graphs other than K2 are (1, 2)-total weight choosable.