Let G be a connected graph on n vertices, and let α,β,γ and δ be edge-disjoint cycles in G such that (i) α,β (respectively, γ,δ) are vertex-disjoint and (ii) ℝαℝ+ℝβℝ=ℝγℝ+ℝδℝ=n, where ℝαℝ denotes the length of α. The authors say that α,β,γ and δ yield two edge-disjoint Hamiltonian cycles by edge exchanges if the four cycles respectively contain edges e,f,g and h such that each of (α-{e})ℝ(β-{f})ℝ{g,h} and (γ-{g})ℝ(δ-{h})ℝ{ℝ{e,f} constitutes a Hamiltonian cycle in G. They show that if G is a nonbipartite, Hamiltonian decomposable graph on an even number of vertices which satisfies certain conditions, then Kronecker product of G and K2 as well as Kronecker product of G and an even cycle admits a Hamiltonian decomposition by means of appropriate edge exchanges among smaller cycles in the product graph.
