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 $〈u_1,u_2,\dots ,u_N,u_1〉$ to emphasize the order of vertices on $C$. Two hamiltonian cycles of $G$, namely $C_1=〈u_1,u_2,\dots ,u_N,u_1〉$ and $C_2=〈v_1,v_2,\dots ,v_N,v_1〉$, are said to be independent if $u_1=v_1$ and $u_i\ne v_i$ for all $2\le i\le N$. A collection of $m$ hamiltonian cycles $C_1,\dots ,C_m$, 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 $\mathcal{I}\mathcal{H}\mathcal{C}\left(G\right)$, 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 $BF\left(n\right)$ denote the $n$-dimensional binary wrapped butterfly graph. Then we prove that $\mathcal{I}\mathcal{H}\mathcal{C}\left(BF\left(n\right)\right)=4$ for all $n\ge 3$.