The construction of mutually independent Hamiltonian cycles in bubble-sort graphs

Abstract

A Hamiltonian cycle C=⟨ u 1, u 2, …, u n(G), u 1 ⟩ with n(G)=number of vertices of G, is a cycle C(u 1; G), where u 1 is the beginning and ending vertex and u i is the ith vertex in C and u i ≠u j for any i≠j, 1≤i, j≤n(G). A set of Hamiltonian cycles {C 1, C 2, …, C k } of G is mutually independent if any two different Hamiltonian cycles are independent. For a hamiltonian graph G, the mutually independent Hamiltonianicity number of G, denoted by h(G), is the maximum integer k such that for any vertex u of G there exist k-mutually independent Hamiltonian cycles of G starting at u. In this paper, we prove that h(B n )=n−1 if n≥4, where B n is the n-dimensional bubble-sort graph.