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 Lwhich assigns to each vertex xa set L(x) of kreal numbers, and assigns to each edge ea set L(e) of k′ real numbers, there is a mapping f: V∪E→ℝ such that f(y)∈L(y) for any y∈V∪Eand for any two adjacent vertices x, 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 K2are (1, 3)‐total weight choosable. Also a graph Gobtained from an arbitrary graph Hby 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 Gobtained from Hby subdividing each edge with at least two vertices is (2, 2)‐total weight choosable as well as (1, 3)‐total weight choosable. © 2010 Wiley Periodicals, Inc. J Graph Theory 66:198‐212, 2011