Embedding Three Edge-Disjoint Hamiltonian Cycles into Locally Twisted Cubes.

The n-dimensional locally twisted cube LTQn, a variation of the hypercube Qn, has the same number of vertices and the same number of edges as Qn, but it has only about half of the diameter of Qn. The existence of the Hamiltonian cycle provides an advantage when implementing algorithms that require a ring structure. In addition, k (≥ 2) edge-disjoint Hamiltonian cycles also provide the edge-fault tolerant Hamiltonicity for the interconnection network. Hung [Theoretical Computer Science 412, 4747–4753, 2011] proved that LTQn with n ⩾ 4 contains two edge-disjoint Hamiltonian cycles, and posted an open problem what is the maximum number of edge-disjoint Hamiltonian cycles in LTQn for n ⩾ 6? In this paper, we show that there exist three edge-disjoint Hamiltonian cycles on LTQn while n ⩾ 6.