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 ∪ 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 4 n 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.