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The planar Tur\'an number of the seven-cycle

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arxiv 2307.06909 v2 pith:6UIKRGPV submitted 2023-07-13 math.CO

Ervin Győri , Alan Li , Runtian Zhou This is my paper · ORCID
classification math.CO
keywords mathcalplanarboundnumbersharpupperconjecturecontain
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abstract

The planar Tur\'an number, $ex_\mathcal{P}(n,H)$, is the maximum number of edges in an $n$-vertex planar graph which does not contain $H$ as a subgraph. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both $ex_\mathcal{P}(n,C_4)$ and $ex_\mathcal{P}(n,C_5)$. Later on, D. Ghosh et al. obtained sharp upper bound of $ex_\mathcal{P}(n,C_6)$ and proposed a conjecture on $ex_\mathcal{P}(n,C_k)$ for $k\geq 7$. In this paper, we give a sharp upper bound $ex_\mathcal{P}(n,C_7)\leq {18\over 7}n-{48\over 7}$, which satisfies the conjecture of D. Ghosh et al. It turns out that this upper bound is also sharp for $ex_\mathcal{P}(n,\{K_4,C_7\})$, the maximum number of edges in an $n$-vertex planar graph which does not contain $K_4$ or $C_7$ as a subgraph.

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Cited by 5 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. An improved upper bound for the planar Tur\'an number of $C_8$

    math.CO 2026-07 conditional novelty 6.0 of 10

    Every n≥8 vertex planar graph with no 8-cycle has at most 69/25 (n−2) edges, improving the previous best coefficient ≈2.99 to 2.76.

  2. Planar Tur\'{a}n numbers of three configurations

    math.CO 2025-09 conditional novelty 6.0 of 10

    Exact planar Turán numbers are established for K1+(P2∪P3), a combined C3/Θ4 configuration, and the disjoint union of C3 and Θ4, with extremal graph characterizations.

  3. Planar Tur\'an number of disjoint union of $C_3$ and $C_5$

    math.CO 2025-07 conditional novelty 6.0 of 10

    For n ≥ 295660, the planar Turán number of C3∪C5 is floor((8n-13)/3), and the unique extremal planar graph is described.

  4. Planar Tur\'an number of two adjacent cycles

    math.CO 2024-11 conditional novelty 6.0 of 10

    The exact planar Turán numbers are determined for the graphs C3-C3 and C3-C4 (two disjoint cycles joined by an edge).

  5. Planar Tur\'an number of quasi-double stars

    math.CO 2025-07 conditional novelty 5.0 of 10

    For quasi-double stars W_{h,k} with 1≤h≤2≤k≤5, the paper proves planar Turán bounds of 3(h+k)/(h+k+2)n for h+k≤5, and two-sided bounds of 5/2 n and 17/6 n for larger cases.

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