Total weight choosability of Cartesian product of graphs

A graph G = (V, E) is called (k, k')-choosable if the following is true: 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 U E -> R such that f(y) epsilon L(y) for any y epsilon V U E and for any two adjacent vertices x, x' Sigma(e epsilon E(x)) f(e) + f(x) not equal Sigma(e epsilon E(x')) f(e) + f(x'). In this paper, we prove that if G is the Cartesian product of an even number of even cycles, or the Cartesian product of an odd number of even cycles and at least one of the cycles has length 4n for some positive integer n, then G is (1, 3)-choosable. In particular, hypercubes of even dimension are (1, 3)-choosable. Moreover, we prove that if G is the Cartesian product of two paths or the Cartesian product of a path and an even cycle, then G is (1, 3)-choosable. In particular, Q is (1, 3)-choosable. (c) 2012 Elsevier Ltd. All rights reserved.