Mutually independent hamiltonian cycles for the pancake graphs and the star graphs

Abstract

A hamiltonian cycle C of a graph G is an ordered set < u(1), u(2,) ..., u(n(G)), u(1)> of vertices such that u(i) not equal u(j) for i not equal j and u(i) is adjacent to u(i+1) for every i is an element of {1, 2, ..., n(G) - 1} and u(n(G)) is adjacent to u(1), where n(G) is the order of G. The vertex u(1) is the starting vertex and u(i) is the ith vertex of C. Two hamiltonian cycles C(1) = < u(1), u(2), ..., u(n(G)), u(1)> and C(2) = < v(1), v(2), ..., v(n(G)), v(1)> of G are independent if u(1) = v(1) and u(i) not equal v(i) for every i is an element of {2, 3, ..., n(G)}. A set of hamiltonian cycles {C(1), C(2), ..., C(k)} of G is mutually independent if its elements are pairwise independent. The mutually independent hamiltonicity IHC(G) of a graph 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, the mutually independent hamiltonicity is considered for two families of Cayley graphs, the n-dimensional pancake graphs P(n) and the n-dimensional star graphs S(n). It is proven that IHC(P(3)) = 1, IHC(P(n)) = n - 1 if n >= 4, IHC(S(n)) = n - 2 if n is an element of {3, 4} and IHC(S(n)) = n - 1 if n >= 5. (C) 2009 Elsevier B.V. All rights reserved.