On the 3-Local Profiles of Graphs

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), ..., p3(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 ≥ 4 . We also give a full description of the triangle-free case, i.e. the intersection of S3 with the hyperplane p3 = 0. This planar domain is characterized by an SDP constraint that is derived from Razborov’s flag algebra theory. C © 2013 Wiley Periodicals, Inc. J. Graph Theory 76: 236–248, 2014