In 1993 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 matchingM, remarkably even ifM 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.
|Journal||Discrete Mathematics and Theoretical Computer Science (Online Edition)|
|Publication status||Published - 2015|
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