Cycles and paths through specified vertices in graphs with a given clique number
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909488114565120 |
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| author | Li, Chengli Xu, Leyou |
| author_facet | Li, Chengli Xu, Leyou |
| contents | B. Bollobás and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence of a cycle through all vertices whose degrees at least $n-ω$ in any $2$-connected graph $G$ of order $n$ with clique number $ω$ unless $G$ is a specific graph. Moreover, we show that for any pair of vertices whose degrees are at least $n-ω+1$ in a graph $G$ of order $n$ with clique number $ω$, there exists a path joining them which contains all vertices of degree at least $n-ω+1$ unless $G$ belongs to certain graph classes. In doing so, we prove the existence of a $(u,v)$-path through all vertices whose degrees at least $\frac{n+1}{2}$ in any graph of order $n$, where $u,v$ are two distinct vertices of degree at least $\frac{n+1}{2}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07534 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cycles and paths through specified vertices in graphs with a given clique number Li, Chengli Xu, Leyou Combinatorics B. Bollobás and G. Brightwell and independently R. Shi proved the existence of a cycle through all vertices whose degrees at least $\frac{n}{2}$ in any $2$-connected graph of order $n$. Motivated by this result, we prove the existence of a cycle through all vertices whose degrees at least $n-ω$ in any $2$-connected graph $G$ of order $n$ with clique number $ω$ unless $G$ is a specific graph. Moreover, we show that for any pair of vertices whose degrees are at least $n-ω+1$ in a graph $G$ of order $n$ with clique number $ω$, there exists a path joining them which contains all vertices of degree at least $n-ω+1$ unless $G$ belongs to certain graph classes. In doing so, we prove the existence of a $(u,v)$-path through all vertices whose degrees at least $\frac{n+1}{2}$ in any graph of order $n$, where $u,v$ are two distinct vertices of degree at least $\frac{n+1}{2}$. |
| title | Cycles and paths through specified vertices in graphs with a given clique number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2502.07534 |