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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.17322 |
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Table of Contents:
- For a family of graphs $\cal F$, a graph $G$ is $\cal F$-free if it does not contain a member of $\cal F$ as a subgraph. The Turán number $\textrm{ex}(n,{\cal F})$ is the maximum number of edges in an $n$-vertex graph which is $\cal F$-free. Let ${\cal C}_{\geq k}$ be the set of cycles with length at least $k$. In this paper, we investigate the Turán number of $\{{\cal C}_{\geq k}, F\}$ for a general graph $F$. To be precise, we determine $\textrm{ex}(n, \{{\cal C}_{\geq k}, F\})$ apart from a constant additive term, where $F$ either is a 2-connected nonbipartite graph or is a 2-connected bipartite graph under some conditions. This is an extension of a previous result on the Turán number of $\{{\cal C}_{\geq k}, K_r\}$ by the first author, Ning, and the third author.