On the Local Profiles of Trees

Bubeck, Sébastien; Linial, Nathan

doi:10.1002/jgt.21865

Cited by 15 publications

(32 citation statements)

References 4 publications

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“…We define the random variable X on E(T ) with uniform distribution that takes the value x e at e ∈ E(T ). Clearly EX = c 3 + o |T | (1). But a 4-vertex tournament is isomorphic to C 4 iff it contains two cyclic triangles with a common edge.…”

Section: B the Minimummentioning

confidence: 99%

“…Let T 1 , T 2 be two families with c 3 (T 1 ) = c 3 (T 2 ) and c 4 (T 1 ) < c < c 4 (T 2 ). We construct an n-vertex tournament T with c 3 (T ) = c 3 (T 1 ) + o n (1) and c 4 (T ) = c + o n (1). Let 0 ≤ p, α ≤ 1 be two constant parameters.…”

Section: Proofsmentioning

confidence: 99%

“…We compute c 3 (1). Choose p such that p(1 − p) = c 3 (T 1 ) = c 3 (T 2 ) and then c 3 (T ) = c 3 (T 1 ) + o (1).…”

Section: Proofsmentioning

confidence: 99%

“…As first observed in [3] the analogous question is answered negatively for graphs. On the other hand, for trees the answer is positive [1]. r To what extent can the lower bound in Lemma 2.5 be improved using higher order flags?…”

Section: Further Directionsmentioning

confidence: 99%

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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: B the Minimummentioning

confidence: 99%

Section: Proofsmentioning

confidence: 99%

“…We compute c 3 (1). Choose p such that p(1 − p) = c 3 (T 1 ) = c 3 (T 2 ) and then c 3 (T ) = c 3 (T 1 ) + o (1).…”

Section: Proofsmentioning

confidence: 99%

Section: Further Directionsmentioning

confidence: 99%

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On the Number of 4-Cycles in a Tournament

Linial

Morgenstern

2015

J. Graph Theory

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“…However, this changes for larger values of k. It is well known that the number of rooted binary trees with k leaves up to isomorphism is the Wedderburn-Etherington number W k . The first few of these numbers are 1,1,1,2,3,6,11,23,46,98,207,451,983, . .…”

Section: Introductionmentioning

confidence: 99%

Inducibility in Binary Trees and Crossings in Random Tanglegrams

Czabarka¹,

Székely²,

Wagner³

2017

SIAM J. Discrete Math.

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Abstract. In analogy to other concepts of a similar nature, we define the inducibility of a rooted binary tree. Given a fixed rooted binary tree B with k leaves, we let γ(B, T ) be the proportion of all subsets of k leaves in T that induce a tree isomorphic to B. The inducibility of B is lim sup |T |→∞ γ(B, T ). We determine the inducibility in some special cases, show that every binary tree has positive inducibility and prove that caterpillars are the only binary trees with inducibility 1. We also formulate some open problems and conjectures on the inducibility. Finally, we present an application to crossing numbers of random tanglegrams.

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