The circumference of a graph with given minimum degree and clique number
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| Format: | Preprint |
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2025
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| _version_ | 1866908620008980480 |
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| author | Chen, Na Tang, Yurui |
| author_facet | Chen, Na Tang, Yurui |
| contents | The circumference denoted by $c(G)$ of a graph $G$ is the length of its longest cycle. Let $δ(G)$ and $ω(G)$ denote the minimum degree and the clique number of a graph $G$, respectively. In [\emph{Electron. J. Combin.} 31(4)(2024) $\#$P4.65], Yuan proved that if $G$ is a 2-connected graph of order $n$, then $c(G)\geq \min\{n,ω(G)+δ(G)\}$ unless $G$ is one of two specific graphs. In this paper, we prove a stability result for the theorem of Erd\H os and Gallai, thereby helping us to characterize all $2$-connected non-hamiltonian graphs whose circumference equals the sum of their clique number and minimum degree. Combining this with Yuan's result, one can deduce that if $G$ is a $2$-connected graph of order $n$, then $c(G)\geq \min\{n,ω(G)+δ(G)+1\}$, unless $G$ belongs to certain specified graph classes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_26233 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The circumference of a graph with given minimum degree and clique number Chen, Na Tang, Yurui Combinatorics The circumference denoted by $c(G)$ of a graph $G$ is the length of its longest cycle. Let $δ(G)$ and $ω(G)$ denote the minimum degree and the clique number of a graph $G$, respectively. In [\emph{Electron. J. Combin.} 31(4)(2024) $\#$P4.65], Yuan proved that if $G$ is a 2-connected graph of order $n$, then $c(G)\geq \min\{n,ω(G)+δ(G)\}$ unless $G$ is one of two specific graphs. In this paper, we prove a stability result for the theorem of Erd\H os and Gallai, thereby helping us to characterize all $2$-connected non-hamiltonian graphs whose circumference equals the sum of their clique number and minimum degree. Combining this with Yuan's result, one can deduce that if $G$ is a $2$-connected graph of order $n$, then $c(G)\geq \min\{n,ω(G)+δ(G)+1\}$, unless $G$ belongs to certain specified graph classes. |
| title | The circumference of a graph with given minimum degree and clique number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.26233 |