Universality of Graphs with Few Triangles and Anti-Triangles

Abstract

We study 3-random-like graphs, that is, sequences of graphs in which the densities of triangles and anti-triangles converge to 1/8. Since the random graph G n,1/2 is, in particular, 3-random-like, this can be viewed as a weak version of quasirandomness. We first show that 3-random-like graphs are 4-universal, that is, they contain induced copies of all 4-vertex graphs. This settles a question of Linial and Morgenstern [9]. We then show that for larger subgraphs, 3-random-like sequences demonstrate a completely different behaviour. We prove that for every graph H on n ≥ R(10, 10) vertices there exist 3-random-like graphs without an induced copy of H. Moreover, we prove that for every ℓ there are 3-random-like graphs which are ℓ-universal but not m-universal when m is sufficiently large compared to ℓ.Research supported by EPSRC grant EP/K033379/1.

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

“…(1) In a follow up paper, Hefetz and Tyomkin [10] settle Problem 1 in the above list and make several additional interesting contributions to this area. (2) We have recently made some progress on Problem 8 above.…”

Frustrated triangles

“…More recently, Linial and Morgenstern [10] showed that every sequence of graphs with t 0 + t 3 asymptotically minimal is 3-universal. Hefetz and Tyomkyn [7] then proved that such sequences are 4-universal, but not necessarily 5-universal, and moreover that any sufficiently large graph H can be avoided by such a sequence.…”