Regularity lemmas for clustering graphs

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.

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