Matchings in Hypercubes Extend to Long Cycles
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866929693146480640 |
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| author | Fink, Jiří Hotmar, Vojtěch |
| author_facet | Fink, Jiří Hotmar, Vojtěch |
| contents | The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional hypercube $Q_n$ can be extended into a Hamilton cycle. We prove that matchings of $Q_n$ containing edges spanning at most $d = 5$ directions can be extended into a Hamilton cycle. We also characterize when these matchings of most $d = 5$ directions can be extended into a Hamilton path between two prescribed vertices. Our proofs work for arbitrary $d$ and $n$ where $d \le n$ assuming some extension properties hold in $Q_d$ which we verified by a computer for $d=5$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_19029 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Matchings in Hypercubes Extend to Long Cycles Fink, Jiří Hotmar, Vojtěch Combinatorics The $n$-dimensional hypercube graph $Q_n$ has as vertices all subsets of $\{1, \ldots, n\}$, and an edge between any two sets that differ in a single element. The Ruskey-Savage conjecture states that every matching of the $n$-dimensional hypercube $Q_n$ can be extended into a Hamilton cycle. We prove that matchings of $Q_n$ containing edges spanning at most $d = 5$ directions can be extended into a Hamilton cycle. We also characterize when these matchings of most $d = 5$ directions can be extended into a Hamilton path between two prescribed vertices. Our proofs work for arbitrary $d$ and $n$ where $d \le n$ assuming some extension properties hold in $Q_d$ which we verified by a computer for $d=5$. |
| title | Matchings in Hypercubes Extend to Long Cycles |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2501.19029 |