Mutually independent hamiltonian cycles of binary wrapped butterfly graphs

Abstract

Effective utilization of communication resources is crucial for improving performance in multiprocessor/communication systems. In this paper, the mutually independent hamiltonicity is addressed for its effective utilization of resources on the binary wrapped butterfly graph. Let G be a graph with N vertices. A hamiltonian cycle C of G is represented by 〈 u 1 , u 2 , … , u N , u 1 〉 to emphasize the order of vertices on C . Two hamiltonian cycles of G , namely C 1 = 〈 u 1 , u 2 , … , u N , u 1 〉 and C 2 = 〈 v 1 , v 2 , … , v N , v 1 〉 , are said to be independent if u 1 = v 1 and u i ≠ v i for all 2 ≤ i ≤ N . A collection of m hamiltonian cycles C 1 , … , C m , starting from the same vertex, are m -mutually independent if any two different hamiltonian cycles are independent. The mutually independent hamiltonicity of a graph G , denoted by I H C ( G ) , is defined to be the maximum integer m such that, for each vertex u of G , there exists a set of m -mutually independent hamiltonian cycles starting from u . Let B F ( n ) denote the n -dimensional binary wrapped butterfly graph. Then we prove that I H C ( B F ( n ) ) = 4 for all n ≥ 3 .