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<p>Given a graph <i>H</i> and an integer <i>p</i> ≥ 2, the edge blow-up graph <i>H</i><sup><i>p</i>+1</sup> of <i>H</i> is the graph obtained by replacing each edge in <i>H</i> with a clique of order <i>p</i> + 1, where the new vertices of the cliques are all distinct. The generalized Turán number ex(<i>n, K</i><sub><i>m</i></sub>, <i>F</i>) denote the maximum number of copies of <i>K</i><sub><i>m</i></sub> in an <i>n</i>-vertex <i>F</i>-free graph. Let <i>C</i><sub><i>t</i></sub> and <i>P</i><sub><i>t</i></sub> denote the cycle and path with <i>t</i> vertices, respectively. In this paper, we obtain the generalized Turán numbers ex(<i>n, K</i><sub><i>m</i></sub>, <i>P</i><Stack> <sub><i>t</i></sub> <sup><i>p</i>+1</sup> </Stack>), ex(<i>n, K</i><sub><i>m</i></sub>, <i>C</i><Stack> <sub><i>t</i></sub> <sup><i>p</i>+1</sup> </Stack>) and characterize the unique graph for <i>P</i><Stack> <sub><i>t</i></sub> <sup><i>p</i>+1</sup> </Stack> and <i>C</i><Stack> <sub><i>t</i></sub> <sup><i>p</i>+1</sup> </Stack> respectively, when <i>t</i> ≥ 3, <i>p</i> ≥ <i>m</i> ≥ 3 and <i>n</i> is sufficiently large.</p>

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Generalized Turán Number for Edge Blow-up of Paths and Cycles

  • 期刊论文

  • 发表日期：

    2026年1月15日

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  • Yuanpei Wang,
  • Liying Kang

摘要

Given a graph H and an integer p ≥ 2, the edge blow-up graph Hp+1 of H is the graph obtained by replacing each edge in H with a clique of order p + 1, where the new vertices of the cliques are all distinct. The generalized Turán number ex(n, Km, F) denote the maximum number of copies of Km in an n-vertex F-free graph. Let Ct and Pt denote the cycle and path with t vertices, respectively. In this paper, we obtain the generalized Turán numbers ex(n, Km, P t p+1 ), ex(n, Km, C t p+1 ) and characterize the unique graph for P t p+1 and C t p+1 respectively, when t ≥ 3, p ≥ m ≥ 3 and n is sufficiently large.