Abstract For a graph G with a positive clustering coefficient C, it is proved that for any positive constant ϵ, the vertex set of G can be partitioned into finitely many parts, say S 1 , S 2 , … , S m , such that all but an ϵ fraction of the triangles in G are contained in the projections of tripartite subgraphs induced by ( S i , S j , S k ) which are ϵ-Δ-regular, where the size m of the partition depends only on ϵ and C. The notion of ϵ-Δ-regular, which is a variation of ϵ-regular for the original regularity lemma, concerns triangle density instead of edge density. Several generalizations and variations of the regularity lemma for clustering graphs are derived.

中文翻译：

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聚类图的正则性引理

摘要 对于具有正聚类系数 C 的图 G，证明对于任意正常数 ϵ，G 的顶点集可以被划分为有限多个部分，例如 S 1 , S 2 , … , S m ，使得所有但是 G 中三角形的一部分 ϵ 包含在由 ( S i , S j , S k ) 引起的三方子图的投影中，这些子图是 ϵ-Δ-正则的，其中分区的大小 m 仅取决于 ϵ 和 C ϵ-Δ-regular 的概念是原始正则引理的 ϵ-regular 的变体，涉及三角形密度而不是边密度。推导出了聚类图的正则性引理的几种概括和变体。