Total Weight Choosability of Graphs

A graph G=(V, E) is called (k, k')-total weight choosable if the following holds: For any total list assignment L which assigns to each vertex x a set L(x) of k real numbers, and assigns to each edge e a set L(e) of k' real numbers, there is a mapping f:V boolean OR E-->R such that f(y)is an element of L(y) for any Y is an element of V boolean OR E and for any two adjacent vertices x, x', Sigma(e is an element of E(x))f(e) + f(x) Sigma(e is an element of E(x'))f(e) + f(x'). We conjecture that every graph is (2, 2)-total weight choosable and every graph without isolated edges is (1,3)-total weight choosable. It follows from results in [7] that complete graphs, complete bipartite graphs, trees other than K-2 are (1, 3)-total weight choosable. Also a graph G obtained from an arbitrary graph H by subdividing each edge with at least three vertices is (1, 3)-total weight choosable. This article proves that complete graphs, trees, generalized theta graphs are (2, 2)-total weight choosable. We also prove that for any graph H, a graph G obtained from H by subdividing each edge with at least two vertices is (2, 2)-total weight choosable as well as (1,3)-total weight choosable. (C) 2010 Wiley Periodicals, Inc. J Graph Theory 66: 198-212, 2011