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Universality of Graphs with Few Triangles and Anti-Triangles

Published online by Cambridge University Press:  29 July 2015

DAN HEFETZ
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK (e-mail: d.hefetz@bham.ac.uk, m.tyomkyn@bham.ac.uk)
MYKHAYLO TYOMKYN
Affiliation:
School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, UK (e-mail: d.hefetz@bham.ac.uk, m.tyomkyn@bham.ac.uk)

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 $\mathcal{G}$ n,1/2 is, in particular, 3-random-like, this can be viewed as a weak version of quasi-randomness. 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 [10]. We then show that for larger subgraphs, 3-random-like sequences demonstrate completely different behaviour. We prove that for every graph H on n ⩾ 13 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.

Type
Paper
Copyright
Copyright © Cambridge University Press 2015 

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