Combinatorics, Probability and Computing
Edge weights and vertex colours
Journal of Combinatorial Theory Series B
Vertex colouring edge partitions
Journal of Combinatorial Theory Series B
Vertex-Colouring Edge-Weightings
Combinatorica
Journal of Graph Theory
Note: Vertex-coloring edge-weightings: Towards the 1-2-3-conjecture
Journal of Combinatorial Theory Series B
Vertex-coloring 2-edge-weighting of graphs
European Journal of Combinatorics
Total weight choosability of graphs
Journal of Graph Theory
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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@?E-R such that f(y)@?L(y) for any y@?V@?E and for any two adjacent vertices x,x^', @?"e"@?"E"("x")f(e)+f(x)@?"e"@?"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"3 is (1,3)-choosable.