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) (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.

The 1,2,3-Conjecture and 1,2-Conjecture for sparse graphs

“…They also proved that every tree with an even number of edges is (1, 2)-weight-choosable. Wong, Yang, and Zhu [11] continued this approach, proving Conjecture 1.4 for graphs with maximum degree 3. Finally, Wong and Zhu [13] proved that every graph is (2, 3)-weight-choosable.…”

“…Wong, Yang, and Zhu [11] proved that the complete multipartite graph K n,m,1,1,...,1 is (2, 2)-weight-choosable and that complete bipartite graphs other than K 2 are (1, 2)-weightchoosable. Bartnicki,Grytczuk,and Niwczyk [4] applied the Combinatorial Nullstellensatz [3] to prove Conjecture 1.3 for complete graphs, complete bipartite graphs, and trees.…”