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On some graph densities in locally dense graphs
Random Structures and Algorithms ( IF 1 ) Pub Date : 2020-11-24 , DOI: 10.1002/rsa.20974
Joonkyung Lee 1
Affiliation  

The Kohayakawa–Nagle–Rödl‐Schacht conjecture roughly states that every sufficiently large locally d‐dense graph G on n vertices must contain at least (1 − o(1))d|E(H )|n|V (H )| copies of a fixed graph H. Despite its important connections to both quasirandomness and Ramsey theory, there are very few examples known to satisfy the conjecture. We provide various new classes of graphs that satisfy the conjecture. First, we prove that adding an edge to a cycle or a tree produces graphs that satisfy the conjecture. Second, we prove that a class of graphs obtained by gluing complete multipartite graphs in a tree‐like way satisfies the conjecture. We also prove an analogous result with odd cycles replacing complete multipartite graphs.

中文翻译:

关于局部密集图中的某些图密度

Kohayakawa–Nagle–Rödl–Schacht猜想粗略地指出,在n个顶点上的每个足够大的局部d密图G必须至少包含(1 −  o(1))d |。EH)| n | V)| 固定图H的副本。尽管它与准随机性和Ramsey理论都有重要关系,但很少有已知的例子可以满足这个猜想。我们提供了满足猜想的各种新型图。首先,我们证明向循环或树添加边会产生满足猜想的图。其次,我们证明通过以树状方式粘合完整的多部分图而获得的一类图满足该猜想。我们还证明了奇数周期替代完整多部分图的相似结果。
更新日期:2021-01-11
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