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Extending a perfect matching to a Hamiltonian cycle

Abstract : 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.
Keywords : Hamiltonian cycle
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Adel Alahmadi, Robert E. L. Aldred, Ahmad Alkenani, Rola Hijazi, P. Solé, et al.. Extending a perfect matching to a Hamiltonian cycle. Discrete Mathematics and Theoretical Computer Science, DMTCS, 2015, Vol. 17 no. 1 (in progress) (1), pp.241--254. ⟨hal-01196849⟩

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