TY - JOUR
T1 - On the 3-local profiles of graphs
AU - Huang, Hao
AU - Linial, Nati
AU - Naves, Humberto
AU - Peled, Yuval
AU - Sudakov, Benny
PY - 2014/7
Y1 - 2014/7
N2 - 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.
AB - 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.
KW - flag algebras
KW - induced densities
KW - local profiles
UR - http://www.scopus.com/inward/record.url?scp=84899476492&partnerID=8YFLogxK
U2 - 10.1002/jgt.21762
DO - 10.1002/jgt.21762
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AN - SCOPUS:84899476492
SN - 0364-9024
VL - 76
SP - 236
EP - 248
JO - Journal of Graph Theory
JF - Journal of Graph Theory
IS - 3
ER -