Many-to-many disjoint paths in hypercubes with faulty vertices

TL;DR

This paper proves that in hypercubes with certain faulty vertices, multiple disjoint paths can be established between two vertex sets, improving previous results in fault-tolerant path connectivity.

Contribution

It establishes new bounds for disjoint paths in hypercubes with faulty vertices, extending known fault-tolerance results.

Findings

Existence of $k$ disjoint fault-free paths under specified fault conditions.

Paths cover at least $2^n - 2f$ vertices, ensuring high connectivity.

Improves upon previous bounds in fault-tolerant hypercube routing.

Abstract

This paper considers the problem of many-to-many disjoint paths in the hypercube $Q_n$ with $f$ faulty vertices and obtains the following result. For any integer $k$ with $1\leq k\leq n-2$, any two sets $S$ and $T$ of $k$ fault-free vertices in different parts of $Q_n\ (n\geq 3)$, if $f\leq 2n-2k-3$ and each fault-free vertex has at least two fault-free neighbors, then there exist $k$ fully disjoint fault-free paths linking $S$ and $T$ which contain at least $2^n-2f$ vertices. This result improves some known results in a sense.