The number of edges in graphs with bounded clique number and circumference
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2024
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| _version_ | 1866915060144668672 |
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| author | Dou, Chunyang Ning, Bo Peng, Xing |
| author_facet | Dou, Chunyang Ning, Bo Peng, Xing |
| contents | Let $\cal H$ be a family of graphs. The Turán number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined ${\rm ex}(n,{\cal H})$ for ${\cal H}=\{K_r,M_k\}$, where $M_k$ is a matching of size $k$. Replacing $M_k$ by $P_k$, Katona and Xiao obtained the Turán number of ${\cal H}=\{K_r,P_k\}$ for $r \leq \lfloor k/2 \rfloor$ and sufficiently large $n$. In addition, they proposed a conjecture for the case of $r \geq \lfloor k/2 \rfloor+1$ and sufficiently large $n$. Motivated by the fact that the result for ${\rm ex}(n,P_k)$ can be deduced from the one for ${\rm ex}(n,{\cal C}_{\geq k})$, we investigate the Turán number of ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$ in this paper. In other words, we aim to determine the maximum number of edges in graphs with clique number at most $r-1$ and circumference at most $k-1$. For ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$, we are able to show the value of ${\rm ex}(n,{\cal H})$ for $r \geq \lfloor (k-1)/2\rfloor+2$ and all $n$. As an application of this result, we confirm Katona and Xiao's conjecture in a stronger form. For $r \leq \lfloor (k-1)/2\rfloor+1$, we manage to show the value of ${\rm ex}(n,{\cal H})$ for sufficiently large $n$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_06449 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The number of edges in graphs with bounded clique number and circumference Dou, Chunyang Ning, Bo Peng, Xing Combinatorics 05C35, 05C38 Let $\cal H$ be a family of graphs. The Turán number ${\rm ex}(n,{\cal H})$ is the maximum possible number of edges in an $n$-vertex graph which does not contain any member of $\cal H$ as a subgraph. As a common generalization of Turán's theorem and Erdős-Gallai theorem on the Turán number of matchings, Alon and Frankl determined ${\rm ex}(n,{\cal H})$ for ${\cal H}=\{K_r,M_k\}$, where $M_k$ is a matching of size $k$. Replacing $M_k$ by $P_k$, Katona and Xiao obtained the Turán number of ${\cal H}=\{K_r,P_k\}$ for $r \leq \lfloor k/2 \rfloor$ and sufficiently large $n$. In addition, they proposed a conjecture for the case of $r \geq \lfloor k/2 \rfloor+1$ and sufficiently large $n$. Motivated by the fact that the result for ${\rm ex}(n,P_k)$ can be deduced from the one for ${\rm ex}(n,{\cal C}_{\geq k})$, we investigate the Turán number of ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$ in this paper. In other words, we aim to determine the maximum number of edges in graphs with clique number at most $r-1$ and circumference at most $k-1$. For ${\cal H}=\{K_r, {\cal C}_{\geq k}\}$, we are able to show the value of ${\rm ex}(n,{\cal H})$ for $r \geq \lfloor (k-1)/2\rfloor+2$ and all $n$. As an application of this result, we confirm Katona and Xiao's conjecture in a stronger form. For $r \leq \lfloor (k-1)/2\rfloor+1$, we manage to show the value of ${\rm ex}(n,{\cal H})$ for sufficiently large $n$. |
| title | The number of edges in graphs with bounded clique number and circumference |
| topic | Combinatorics 05C35, 05C38 |
| url | https://arxiv.org/abs/2410.06449 |