Matchings in hypercubes extend to long cycles

2024/01/03 by Jiří Fink, Torsten Mütze, Fink, Jiří +1
Computer Science · #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Interconnection Networks and Systems #Optimization and Search Problems

paper · pdf · doi:10.48550/arxiv.2401.01769

openalex publication_date 2024/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

The d-dimensional hypercube graph Q_d has as vertices all subsets of \1,…,d\, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture asserts that every matching of Q_d, d≥ 2, can be extended to a Hamilton cycle, i.e., to a cycle that visits every vertex exactly once. We prove that every matching of Q_d, d≥ 2, can be extended to a cycle that visits at least a 2/3-fraction of all vertices.

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