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 u, Σe∈E(u) f(e)+f(u) ≠ Σe∈E(v)/(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 K2 are (1, 2)-total weight choosable.