TY - JOUR

T1 - Universality of Graphs with Few Triangles and Anti-Triangles

AU - Hefetz, Dan

AU - Tyomkyn, Mykhaylo

N1 - Publisher Copyright:
© Cambridge University Press 2015.

PY - 2016/7/1

Y1 - 2016/7/1

N2 - 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 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 â"" there are 3-random-like graphs which are â""-universal but not m-universal when m is sufficiently large compared to â"".

AB - 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 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 â"" there are 3-random-like graphs which are â""-universal but not m-universal when m is sufficiently large compared to â"".

UR - http://www.scopus.com/inward/record.url?scp=84938395509&partnerID=8YFLogxK

U2 - 10.1017/S0963548315000188

DO - 10.1017/S0963548315000188

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AN - SCOPUS:84938395509

SN - 0963-5483

VL - 25

SP - 560

EP - 576

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 4

ER -