Hamiltonian paths extending a set of matchings in hypercubes

TL;DR

This paper investigates conditions under which a hypercube contains a Hamiltonian path that extends a given matching set, generalizing previous results and establishing new bounds for such paths in high-dimensional hypercubes.

Contribution

It extends prior work by proving the existence of Hamiltonian paths in hypercubes that include a matching set of edges, for larger matchings than previously known.

Findings

Hamiltonian paths exist in hypercubes with matchings up to size 3n - 13 for n ≥ 5.

The paper generalizes previous results on Hamiltonian paths with specific vertex pairs.

It establishes new bounds on the size of matchings that can be extended to Hamiltonian paths.

Abstract

The hypercube \( Q_n \) contains a Hamiltonian path joining \( x \) and \( y \) (where $x$ and $y$ from the opposite partite set) containing \( P \) if and only if the induced subgraph of \( P \) is a linear forest, where none of these paths have \( x \) or \( y \) as internal vertices nor both as endpoints. Dvo\v{r}\'ak and Gregor answered a problem posed by Caha and Koubek and proved that for every \( n \geq 5 \), there exist vertices \( x \) and \( y \) with a set of \( 2n - 4 \) edges in \( Q_n \) that extend to the Hamiltonian path joining \( x \) and \( y \). This paper examines the Hamiltonian properties of hypercubes with a matching set. Let consider the hypercube \( Q_n \), for \( n \geq 5 \) and a set of matching \( M \) such that \( |M| \leq 3n - 13 \). We prove a Hamiltonian path exists joining two vertices $x$ and $y$ in \( Q_n \) from opposite partite sets containing $M$.