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A note on 3‐partite graphs without 4‐cycles
Journal of Combinatorial Designs ( IF 0.5 ) Pub Date : 2020-06-22 , DOI: 10.1002/jcd.21742
Zequn Lv 1 , Mei Lu 1 , Chunqiu Fang 1
Affiliation  

Let C 4 be a cycle of order 4. Write e x ( n , n , n , C 4 ) for the maximum number of edges in a balanced 3‐partite graph whose vertex set consists of three parts, each has n vertices that have no subgraph isomorphic to C 4 . In this paper, we show that e x ( n , n , n , C 4 ) 3 2 n ( p + 1 ) , where n = p ( p 1 ) 2 and p is a prime number. Note that e x ( n , n , n , C 4 ) ( 3 2 2 + o ( 1 ) ) n 3 2 from Tait and Timmons's works. Since for every integer m , one can find a prime p such that m p ( 1 + o ( 1 ) ) m , we obtain that lim n e x ( n , n , n , C 4 ) 3 2 2 n 3 2 = 1 .

中文翻译:

关于不包含4个循环的3个部分图的注释

C 4 是订单的一个周期4.写 Ë X ñ ñ ñ C 4 平衡的三部分图中顶点集由三部分组成的最大边数 ñ 没有子图同构的顶点 C 4 。在本文中,我们表明 Ë X ñ ñ ñ C 4 3 2 ñ p + 1个 ,在哪里 ñ = p p - 1个 2 p 是素数。注意 Ë X ñ ñ ñ C 4 3 2 2 + Ø 1个 ñ 3 2 摘自Tait和Timmons的作品。因为对于每个整数 ,可以找到素数 p 这样 p 1个 + Ø 1个 ,我们得到 ñ Ë X ñ ñ ñ C 4 3 2 2 ñ 3 2 = 1个
更新日期:2020-06-22
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