Weight Choosability of Graphs

Suppose the edges of a graph G are assigned 3-element lists of real weights. Is it possible to choose a weight for each edge from its list so that the sums of weights around adjacent vertices were different? We prove that the answer is positive for several classes of graphs, including complete graphs, complete bipartite graphs, and trees (except K(2)). The argument is algebraic and uses permanents of matrices and Combinatorlal Nullstellensatz. We also consider a directed version of the problem. We prove by an elementary argument that for digraphs the answer to the above question is positive even with lists of size two. (c) 2008 Wiley Periodicals, Inc. J Graph Theory 60: 242-256, 2009