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 ≠ u j for i ≠ j and u i is adjacent to u i + 1 for every i ∈ { 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 i th 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 ≠ v i for every i ∈ { 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 I H C ( 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 I H C ( P 3 ) = 1 , I H C ( P n ) = n − 1 if n ≥ 4 , I H C ( S n ) = n − 2 if n ∈ { 3 , 4 } and I H C ( S n ) = n − 1 if n ≥ 5 .