Hamiltonicity of Schrijver graphs and stable Kneser graphs
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866914817487405056 |
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| author | Mütze, Torsten Namrata |
| author_facet | Mütze, Torsten Namrata |
| contents | For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers $k\geq 1$, $s\geq 2$, and $n \geq sk+1$, the $s$-stable Kneser graph $S(n,k,s)$ has as vertices all $k$-element subsets of $[n]$ in which any two elements are in cyclical distance at least $s$. We prove that all the graphs $S(n,k,s)$, in particular Schrijver graphs $S(n,k)=S(n,k,2)$, admit a Hamilton cycle that can be computed in time $\mathcal{O}(n)$ per generated vertex. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01681 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hamiltonicity of Schrijver graphs and stable Kneser graphs Mütze, Torsten Namrata Combinatorics Discrete Mathematics For integers $k\geq 1$ and $n\geq 2k+1$, the Schrijver graph $S(n,k)$ has as vertices all $k$-element subsets of $[n]:=\{1,2,\ldots,n\}$ that contain no two cyclically adjacent elements, and an edge between any two disjoint sets. More generally, for integers $k\geq 1$, $s\geq 2$, and $n \geq sk+1$, the $s$-stable Kneser graph $S(n,k,s)$ has as vertices all $k$-element subsets of $[n]$ in which any two elements are in cyclical distance at least $s$. We prove that all the graphs $S(n,k,s)$, in particular Schrijver graphs $S(n,k)=S(n,k,2)$, admit a Hamilton cycle that can be computed in time $\mathcal{O}(n)$ per generated vertex. |
| title | Hamiltonicity of Schrijver graphs and stable Kneser graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2401.01681 |