Planar Turán Numbers of Cubic Graphs and Disjoint Union of Cycles

Pub Date : 2024-02-25 DOI:10.1007/s00373-024-02750-3

引用次数: 0

Abstract

The planar Turán number of a graph H, denoted by \(ex_{_\mathcal {P}}(n,H)\) , is the maximum number of edges in a planar graph on n vertices without containing H as a subgraph. This notion was introduced by Dowden in 2016 and has attracted quite some attention since then; those work mainly focus on finding \(ex_{_\mathcal {P}}(n,H)\) when H is a cycle or Theta graph or H has maximum degree at least four. In this paper, we first completely determine the exact values of \(ex_{_\mathcal {P}}(n,H)\) when H is a cubic graph. We then prove that \(ex_{_\mathcal {P}}(n,2C_3)=\lceil 5n/2\rceil -5\) for all \(n\ge 6\) , and obtain the lower bounds of \(ex_{_\mathcal {P}}(n,2C_k)\) for all \(n\ge 2k\ge 8\) . Finally, we also completely determine the exact values of \(ex_{_\mathcal {P}}(n,K_{2,t})\) for all \(t\ge 3\) and \(n\ge t+2\) .

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立体图形的平面图兰数和循环的不相邻联盟

摘要图 H 的平面图兰数，用 \(ex_{_\mathcal {P}}(n,H)\) 表示。表示，是 n 个顶点上的平面图中不包含 H 作为子图的最大边数。这一概念由 Dowden 于 2016 年提出，此后引起了相当多的关注；这些工作主要集中在寻找当 H 是一个循环图或 Theta 图或 H 的最大度至少为四时的 \(ex_{_\mathcal {P}}(n,H)\) 。在本文中，我们首先完全确定了当 H 是立方图时 \(ex_{_\mathcal {P}}(n,H)\) 的精确值。然后我们证明了 \(ex_{_\mathcal {P}}(n,2C_3)=\lceil 5n/2\rceil -5\) for all \(n\ge 6\) ，并得到了 \(ex_{_\mathcal {P}}(n,2C_k)\) for all \(n\ge 2k\ge 8\) 的下界。最后，我们还完全确定了所有（tge 3\) 和（nge t+2\) 的 \(ex_{_\mathcal {P}}(n,K_{2,t})\) 的精确值。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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