On graphs without cycles of length 1 modulo 3
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arXiv
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| Autori principali: | , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915183133196288 |
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| author | Bai, Yandong Li, Binlong Pan, Yufeng Zhang, Shenggui |
| author_facet | Bai, Yandong Li, Binlong Pan, Yufeng Zhang, Shenggui |
| contents | Burr and Erdős conjectured in 1976 that for every two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains an even integer, an $n$-vertex graph containing no cycles of length $\ell$ modulo $k$ can contain at most a linear number of edges on $n$. Bollobás confirmed this conjecture in 1977 and then Erdős proposed the problem of determining the exact value of the maximum number of edges in such a graph. For the above $k$ and $\ell$, define $c_{\ell,k}$ to be the least constant such that every $n$-vertex graph with at least $c_{\ell,k}\cdot n$ edges contains a cycle of length $\ell$ modulo $k$. The precise (or asymptotic) values of $c_{\ell,k}$ are known for very few pairs $\ell$ and $k$. In this paper, we precisely determine the maximum number of edges in a graph containing no cycles of length 1 modulo 3. In particular, we show that every $n$-vertex graph with at least $\frac{5}{3}(n-1)$ edges contains a cycle of length 1 modulo 3, unless $9|(n-1)$ and each block of the graph is a Petersen graph. As a corollary, we obtain that $c_{1,3}=\frac{5}{3}$. This is the last remaining class modulo $k$ for $1\leqslant k\leqslant 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_03504 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On graphs without cycles of length 1 modulo 3 Bai, Yandong Li, Binlong Pan, Yufeng Zhang, Shenggui Combinatorics Burr and Erdős conjectured in 1976 that for every two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains an even integer, an $n$-vertex graph containing no cycles of length $\ell$ modulo $k$ can contain at most a linear number of edges on $n$. Bollobás confirmed this conjecture in 1977 and then Erdős proposed the problem of determining the exact value of the maximum number of edges in such a graph. For the above $k$ and $\ell$, define $c_{\ell,k}$ to be the least constant such that every $n$-vertex graph with at least $c_{\ell,k}\cdot n$ edges contains a cycle of length $\ell$ modulo $k$. The precise (or asymptotic) values of $c_{\ell,k}$ are known for very few pairs $\ell$ and $k$. In this paper, we precisely determine the maximum number of edges in a graph containing no cycles of length 1 modulo 3. In particular, we show that every $n$-vertex graph with at least $\frac{5}{3}(n-1)$ edges contains a cycle of length 1 modulo 3, unless $9|(n-1)$ and each block of the graph is a Petersen graph. As a corollary, we obtain that $c_{1,3}=\frac{5}{3}$. This is the last remaining class modulo $k$ for $1\leqslant k\leqslant 4$. |
| title | On graphs without cycles of length 1 modulo 3 |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2503.03504 |