Abstract
The Turán number ex(n, H) is the maximum number of edges in any graph of order n that contains no copy of H as a subgraph. For any three positive integers p, q, r with \(p\le q\le r\) and \(q\ge 2\), let \(\theta (p,q,r)\) denote the graph obtained from three internally disjoint paths with the same pair of endpoints, where the three paths are of lengths p, q, r, respectively. Let \(k=p+q+r-1\). In this paper, we obtain the exact value of \(ex(n,\theta (p,q,r))\) and characterize the unique extremal graph for \(n\ge 9k^2-3k\) and any p, q, r with different parities. This extends a known result on odd cycles.
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Acknowledgements
The authors are grateful to the referees for their valuable comments and suggestions. The research of Mingqing Zhai is supported by NSFC (No. 11971445).
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Zhai, M., Fang, L. & Shu, J. On the Turán Number of Theta Graphs. Graphs and Combinatorics 37, 2155–2165 (2021). https://doi.org/10.1007/s00373-021-02342-5
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DOI: https://doi.org/10.1007/s00373-021-02342-5