Directed Hamilton Cycle Decompositions of the Tensor Products of Symmetric Digraphs

Abstract

In this paper, the existence of directed Hamilton cycle decompositions of symmetric digraphs of tensor products of regular graphs, namely, $(K_r \times K_s)^*,\,\,((K_r \circ \overline{K}_s) \times K_n)^*,\,\,((K_r \times K_s) \times K_m)^*,\,\,((K_r \circ \overline{K}_s) \times (K_m \circ \overline{K}_n))^*$ and $(K_{r,r} \times (K_m \circ \overline{K}_n))^*$, where × and ∘ 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 Szász [6], see also Alspach et al. [1]: If D ₁ and D ₂ are directed Hamilton cycle decomposable digraphs, then D ₁ ∘ D ₂ is directed Hamilton cycle decomposable. Ng [17] also has proved that the complete symmetric r-partite regular digraph, $K_{r(s)}^{*} = (K_r \circ \overline{K}_s)^*$, is decomposable into directed Hamilton cycles if and only if $(r,s) \ne (4,1)$ or (6, 1); using the results obtained here, we give a short proof of it, when $r \notin {4,6}$.