Counting triangles in regular graphs
Fuente:
arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866916095807455232 |
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| author | He, Jialin Hou, Xinmin Ma, Jie Xie, Tianying |
| author_facet | He, Jialin Hou, Xinmin Ma, Jie Xie, Tianying |
| contents | In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andrásfai-Erdős-Sós Theorem has established that $t(n,k)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\frac{2n}{5}+\frac{12\sqrt{n}}{5}<k<\frac{n}{2}$. Here, we bridge the gap between the aforementioned results by determining the precise value of $t(n,k)$ in the entire range $\frac{2n}{5}<k<\frac{n}{2}$. This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large $n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_02993 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Counting triangles in regular graphs He, Jialin Hou, Xinmin Ma, Jie Xie, Tianying Combinatorics In this paper, we investigate the minimum number of triangles, denoted by $t(n,k)$, in $n$-vertex $k$-regular graphs, where $n$ is an odd integer and $k$ is an even integer. The well-known Andrásfai-Erdős-Sós Theorem has established that $t(n,k)>0$ if $k>\frac{2n}{5}$. In a striking work, Lo has provided the exact value of $t(n,k)$ for sufficiently large $n$, given that $\frac{2n}{5}+\frac{12\sqrt{n}}{5}<k<\frac{n}{2}$. Here, we bridge the gap between the aforementioned results by determining the precise value of $t(n,k)$ in the entire range $\frac{2n}{5}<k<\frac{n}{2}$. This confirms a conjecture of Cambie, de Joannis de Verclos, and Kang for sufficiently large $n$. |
| title | Counting triangles in regular graphs |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2309.02993 |