Prescribed matchings extend to Hamiltonian cycles in hypercubes with faulty edges

TL;DR

This paper investigates the extension of matchings to Hamiltonian cycles in hypercubes with faulty edges, proving that most matchings can be extended even with some edge faults, advancing understanding of hypercube connectivity.

Contribution

It extends previous results by showing matchings of size up to 2n-1 can be extended to Hamiltonian cycles in faulty hypercubes, with specific fault tolerance bounds.

Findings

Matchings of up to 2n-1 edges extend to Hamiltonian cycles in hypercubes.

Extension remains possible with up to n-1 - ceiling(|M|/2) faulty edges for n≥4.

The result holds except for one specific exception.

Abstract

Ruskey and Savage asked the following question: Does every matching of $Q_{n}$ for $n\geq2$ extend to a Hamiltonian cycle of $Q_{n}$? J. Fink showed that the question is true for every perfect matching, and solved the Kreweras' conjecture. In this paper we consider the question in hypercubes with faulty edges. We show that every matching $M$ of at most $2n-1$ edges can be extended to a Hamiltonian cycle of $Q_{n}$ for $n\geq2$. Moreover, we can prove that when $n\geq4$ and $M$ is nonempty this result still holds even if $Q_{n}$ has at most $n-1-\lceil\frac{|M|}{2}\rceil$ faulty edges with one exception.