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On the 3-local profiles of graphs. (English) Zbl 1305.05120
Authors’ abstract: For a graph \(G\), let \(p_i(G)\), \(i=0,\ldots,3\) be the probability that three distinct random vertices span exactly \(i\) edges. We call \((p_0(G), \ldots,p_3(G))\) the 3-local profile of \(G\). We investigate the set \(\mathcal{S}_3 \subset \mathbb{R}^4\) of all vectors \((p_0,\ldots,p_3)\) that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of \(\mathcal{S}_3\) 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 \(p_0+p_3 \geq \frac{1}{4}\). We also give a full description of the triangle-free case, i.e. the intersection of \(\mathcal{S}_3\) with the hyperplane \(p_3=0\). This planar domain is characterized by an SDP constraint that is derived from Razborov’s flag algebra theory.

MSC:
05C42 Density (toughness, etc.)
05C75 Structural characterization of families of graphs
05C80 Random graphs (graph-theoretic aspects)
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