On the 3-local profiles of graphs

Hao Huang, Nati Linial, Humberto Naves, Yuval Peled, Benny Sudakov

Research output: Contribution to journalArticlepeer-review

12 Scopus citations


For a graph G, let pi(G),i=0,.,3 be the probability that three distinct random vertices span exactly i edges. We call (p0(G),.,p 3(G)) the 3-local profile of G. We investigate the set S3'R4 of all vectors (p0,.,p3) that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of S3 to the (p0,p3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p0+p3≥14. We also give a full description of the triangle-free case, i.e. the intersection of S3 with the hyperplane p 3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

Original languageAmerican English
Pages (from-to)236-248
Number of pages13
JournalJournal of Graph Theory
Issue number3
StatePublished - Jul 2014


  • flag algebras
  • induced densities
  • local profiles


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