Adel Alahmadi ; Robert E. L. Aldred ; Ahmad Alkenani ; Rola Hijazi ; P. Solé et al. - Extending a perfect matching to a Hamiltonian cycle

dmtcs:2105 - Discrete Mathematics & Theoretical Computer Science, March 28, 2015, Vol. 17 no. 1 - https://doi.org/10.46298/dmtcs.2105
Extending a perfect matching to a Hamiltonian cycleArticle

Authors: Adel Alahmadi 1; Robert E. L. Aldred 2; Ahmad Alkenani 1; Rola Hijazi 1; P. Solé 3,4; Carsten Thomassen ORCID5,1

  • 1 Department of Mathematics [Jeddah]
  • 2 Department of Mathematics and Statistics [Dunedin]
  • 3 Department of Mathematics [Jeddah]
  • 4 Télécom ParisTech
  • 5 Department of Applied Mathematics and Computer Science [Lyngby]

Ruskey and Savage conjectured that in the d-dimensional hypercube, every matching M can be extended to a Hamiltonian cycle. Fink verified this for every perfect matching M, remarkably even if M contains external edges. We prove that this property also holds for sparse spanning regular subgraphs of the cubes: for every d ≥7 and every k, where 7 ≤k ≤d, the d-dimensional hypercube contains a k-regular spanning subgraph such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle. We do not know if this result can be extended to k=4,5,6. It cannot be extended to k=3. Indeed, there are only three 3-regular graphs such that every perfect matching (possibly with external edges) can be extended to a Hamiltonian cycle, namely the complete graph on 4 vertices, the complete bipartite 3-regular graph on 6 vertices and the 3-cube on 8 vertices. Also, we do not know if there are graphs of girth at least 5 with this matching-extendability property.


Volume: Vol. 17 no. 1
Section: Graph Theory
Published on: March 28, 2015
Submitted on: May 21, 2014
Keywords: Hamiltonian cycle,[INFO.INFO-DM] Computer Science [cs]/Discrete Mathematics [cs.DM],[INFO.INFO-HC] Computer Science [cs]/Human-Computer Interaction [cs.HC]

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