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arXiv · 1211.3106

On the 3-local profiles of graphs

Hao Huang·Nati Linial·Humberto Naves·Yuval Peled·Benny Sudakov
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Abstract

For a graph G, let p_i(G), i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p_0(G),...,p_3(G)) the 3-local profile of G. We investigate the set ${\cal S}_3 \subset \mathbb R^4$ of all vectors (p_0,...,p_3) that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of ${\cal S}_3$ to the (p_0, p_3) 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 p_0+p_3\geq 1/4. We also give a full description of the triangle-free case, i.e., the intersection of ${\cal S}_3$ with the hyperplane p_3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

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Hao Huang, Nati Linial, Humberto Naves, Yuval Peled, Benny Sudakov. 2013-12-08. On the 3-local profiles of graphs. https://arxiv.org/abs/1211.3106

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