Unpaired many-to-many disjoint path cover of balanced hypercubes

TL;DR

This paper proves the existence of a large unpaired disjoint path cover in balanced hypercubes, improving known results and establishing the optimal upper bound for the number of disjoint paths.

Contribution

It establishes the existence of an unpaired (2n-2)-disjoint path cover in balanced hypercubes, which is an improvement over previous paired path cover results and proves the bound is optimal.

Findings

Existence of unpaired (2n-2)-disjoint path cover in BH_n

Improved upon known paired path cover results

Proved the upper bound of 2n-2 is best possible

Abstract

The balanced hypercube $BH_n$, a variant of the hypercube, was proposed as a desired interconnection network topology. It is known that $BH_n$ is bipartite. Assume that $S=\{s_1,s_2,\cdots,s_{2n-2}\}$ and $T=\{t_1,t_2,\cdots,t_{2n-2}\}$ are any two sets of vertices in different partite sets of $BH_n$ ($n\geq2$). It has been proved that there exists paired 2-disjoint path cover of $BH_n$. In this paper, we prove that there exists unpaired $(2n-2)$-disjoint path cover of $BH_n$ ($n\geq2$) from $S$ to $T$, which improved some known results. The upper bound $2n-2$ of the number of disjoint paths in unpaired $(2n-2)$-disjoint path cover is best possible.