Directed Hamilton Cycle Decompositions of the Tensor Products of Symmetric Digraphs

In this paper, the existence of directed Hamilton cycle decompositions of symmetric digraphs of tensor products of regular graphs, namely, (K-r x K-s)*, ((K-r o (K) over bar (s) x K-n)*, ((K-r x K-s) x K-m)*, ((K-r o (K) over bar (s)) x (K-m o (K) over bar (n)))* and (K-r,(r) x (K-m o (K) over bar (n)))*, where x and o denote the tensor product and the wreath product of graphs, respectively, are proved. In [16], Ng has obtained a partial solution to the following conjecture of Baranyai and Szasz [6], see also Alspach et al. [1]: If D-1 and D-2 are directed Hamilton cycle decomposable digraphs, then D-1 o D-2 is directed Hamilton cycle decomposable. Ng [17] also has proved that the complete symmetric r-partite regular digraph, K*(r(s)) = (K-r o (K) over bar (s))*, is decomposable into directed Hamilton cycles if and only if (r, s) not equivalent to (4, 1) or (6, 1); using the results obtained here, we give a short proof of it, when r is not an element of {4, 6}.