On the 3‐Local Profiles of Graphs

Abstract

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 scriptS3⊂double-struckR4 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 p3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

Graphs with Few 3‐Cliques and 3‐Anticliques are 3‐Universal

“…It is not difficult to see that this question is equivalent to the problem of minimizing given , which is analogous to an interesting question about graphs: Let , given how small can be? (This question is stated in its general form in though it was probably posed earlier.) Razborov's recent solution for was a major achievement in local graph theory.…”

“…In the present article we add some piece to what is known about 3 . At this writing even 3 is not yet fully understood (but see [11,16]). The state of our knowledge of l for l ≥ 4 is really very limited, though some work already exists, e.g., [7,8,12,13,[17][18][19].…”

On the Local Profiles of Trees

“…Not only is nonconvex, it is even computationally infeasible to derive a description of its convex hull (see ). Our understanding of the sets is rather fragmentary (e.g., ). Flag algebras are a major tool in such investigations.…”

Permutations with fixed pattern densities

“…The feasible region for fixed densities ρ 123 versus ρ 321 is the same as the feasible region for triangle density x = d ( K 3 , G ) versus anti‐triangle density of graphons . Let C be the line segment for , D the x ‐axis from to x = 1, and E the y ‐axis from to y = 1.…”

“…To see that is also the feasible region for 123 versus 321 density of permutons, an argument much like the one above for 12 versus 123 can be (and was, by ) given. Permutons realizing various boundary points are illustrated in Figure ; they correspond to the extremal graphons described in . The rest are filled in by parameterization and a topological argument (essentially Theorem above, once we add an extra spike poking into R from the dimple to continuously interpolate between the two permutons at the dimple).…”