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

Linial, Nathan; Morgenstern, Avraham

doi:10.1002/jgt.21801

Cited by 5 publications

(16 citation statements)

References 26 publications

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“…In [8] we initiated the study of 4-local profiles of tournaments, namely the set P = {(t 4 (T ), c 4 (T ), w(T ), l(T ))|T is a family of tournaments for which all the limits exist} ⊆ R 4 .…”

Section: A Notationmentioning

confidence: 99%

On the Number of 4-Cycles in a Tournament

Linial

Morgenstern

2015

J. Graph Theory

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If T is an n-vertex tournament with a given number of 3-cycles, what can be said about the number of its 4-cycles? The most interesting range of this problem is where T is assumed to have c ⋅ n 3 cyclic triples for some c > 0 and we seek to minimize the number of 4-cycles. We conjecture that the (asymptotic) minimizing T is a random blow-up of a constant-sized transitive tournament. Using the method of flag algebras, we derive a lower bound that almost matches the conjectured value. We are able to answer the easier problem of maximizing the number of 4-cycles. These questions can be equivalently stated in terms of transitive subtournaments. Namely, given the number of transitive triples in T , how many transitive quadruples can it have? As far as we know, this is the first study of inducibility in tournaments.

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Section: A Notationmentioning

confidence: 99%

On the Number of 4-Cycles in a Tournament

Linial

Morgenstern

2015

J. Graph Theory

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“…Note that our definition of -universal sequences is slightly different from the one given in [10]. The latter required additionally that p H (G k ) be bounded away from 0 for each H of order .…”

Section: Question 12 ([10]mentioning

confidence: 99%

“…The next lemma and its corollary can be viewed as a strengthening of the 3-universality result from [10] (although, unlike Linial and Morgenstern, we do not optimize the error term ε). It provides additional information about the 3-local profile of Goodman graphs, asserting that it is determined completely by p 0 (and, equivalently, by p 3 ).…”

Section: Preliminariesmentioning

confidence: 99%

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Universality of Graphs with Few Triangles and Anti-Triangles

Hefetz¹,

Tyomkyn²

2015

Combinator. Probab. Comp.

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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.

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“…Goodman [6] also conjectured the maximum of t 0 + t 3 amongst the graphs with a given number of edges, which was later proved by Olpp [13]. 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.…”

Section: Introductionmentioning

confidence: 95%

Frustrated triangles

Kittipassorn

Mészáros²

2015

Discrete Mathematics

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a b s t r a c tA triple of vertices in a graph is a frustrated triangle if it induces an odd number of edges.We study the set F n ⊂ [0,  n 3  ] of possible number of frustrated triangles f (G) in a graph G on n vertices. We prove that about two thirds of the numbers in [0, n 3/2 ] cannot appear in F n , and we characterise the graphs G with f (G) ∈ [0, n 3/2]. More precisely, our main result is that, for each n ≥ 3, F n contains two interlacing sequences 0and only if G can be obtained from a complete bipartite graph by flipping exactly t edges/nonedges. On the other hand, we show, for all n sufficiently large, that if m, then m ∈ F n where f (n) is asymptotically best possible with f (n) ∼ n 3/2 for n even and f (n) ∼ √ 2n 3/2 for n odd. Furthermore, we determine the graphs with the minimum number of frustrated triangles amongst those with n vertices and e ≤ n 2 /4 edges.

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

Cited by 5 publications

References 26 publications

On the Number of 4-Cycles in a Tournament

On the Number of 4-Cycles in a Tournament

Universality of Graphs with Few Triangles and Anti-Triangles

Frustrated triangles

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