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Supersaturation of even linear cycles in linear hypergraphs
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2020-06-23 , DOI: 10.1017/s0963548320000206 Tao Jiang , Liana Yepremyan
Combinatorics, Probability and Computing ( IF 0.9 ) Pub Date : 2020-06-23 , DOI: 10.1017/s0963548320000206 Tao Jiang , Liana Yepremyan
A classical result of Erdős and, independently, of Bondy and Simonovits [3] says that the maximum number of edges in an n -vertex graph not containing C2k , the cycle of length 2k , is O (n 1+1/k ). Simonovits established a corresponding supersaturation result for C 2k ’s, showing that there exist positive constants C ,c depending only on k such that every n -vertex graph G with e (G )⩾ Cn 1+1/k contains at least c (e (G )/v (G ))2k copies of C 2k , this number of copies tightly achieved by the random graph (up to a multiplicative constant).In this paper we extend Simonovits' result to a supersaturation result of r -uniform linear cycles of even length in r -uniform linear hypergraphs. Our proof is self-contained and includes the r = 2 case. As an auxiliary tool, we develop a reduction lemma from general host graphs to almost-regular host graphs that can be used for other supersaturation problems, and may therefore be of independent interest.
中文翻译:
线性超图中偶数线性循环的过饱和
Erdős 以及 Bondy 和 Simonovits [3] 的经典结果表明,n -顶点图不包含C2k , 长度为 2 的循环ķ , 是○ (n 1+1/ķ )。Simonovits 建立了相应的过饱和结果C 2ķ , 表明存在正常数C ,C 只取决于ķ 这样每一个n -顶点图G 和e (G )⩾cn 1+1/ķ 至少包含C (e (G )/v (G ))2ķ 的副本C 2ķ ,这个副本数是由随机图紧密实现的(高达一个乘法常数)。在本文中,我们将 Simonovits 的结果扩展到过饱和结果r - 均匀长度的线性循环r - 均匀线性超图。我们的证明是独立的,包括r = 2 例。作为辅助工具,我们开发了从一般宿主图到几乎规则宿主图的归约引理,可用于其他过饱和问题,因此可能具有独立的兴趣。
更新日期:2020-06-23
中文翻译:
线性超图中偶数线性循环的过饱和
Erdős 以及 Bondy 和 Simonovits [3] 的经典结果表明,