A note on Hamiltonian decomposition of Bubble-Sort graphs

摘要

The Bubble-Sort graph, denoted by B-n (n is positive integer), is a special class of Cayley graph model. In 2009, Shi and Niu [Hamiltonian decomposition of some interconnection networks, in Combinatorial Optimization and Applications, D.-Z. Du, X. Hu, and P.M. Pardalos, eds., Springer, Huangshan, 2009, pp. 231-237.] proposed the following conjecture: (i) If n is odd then B-n is the union of (n - 1)/2 edge-disjoint Hamiltonian cycles. (ii) If n is even then B-n is the union of (n - 2)/2 edge-disjoint Hamiltonian cycles and a perfect matching. In this paper, we give a construction of the decomposition of Bubble-Sort graph Bn+1 with n odd using the decomposition of B-n. Moreover, if the decomposition of B-n is given using the decomposition of Bn-1 then the conjecture is proved.