On claws belonging to every tournament

Lu, Xiaoyun

doi:10.1007/bf01206360

Cited by 11 publications

(7 citation statements)

References 4 publications

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“…By Chebyshev's inequality, (12) and (14) it follows that P X − E(X) > ε 2 · E(X) ≤ Var(X) εE(X)/2 2 = o(1), which together with (12) proves the lemma.…”

Section: Most Oriented Trees Are Nicementioning

confidence: 62%

“…Indeed, Saks and Sós conjectured that every claw on n vertices with maximum degree at most

n / 2

is unavoidable. Lu gave a counterexample to this conjecture, but in the other direction showed that every claw with maximum degree at most

3 n / 8

is unavoidable. Lu, Wang, and Wong then extended these results by showing that every claw with maximum degree at most

19 n / 50

is unavoidable, but that there exist claws with maximum degree approaching

11 n / 23

which are avoidable.…”

Section: Introductionmentioning

confidence: 99%

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Unavoidable trees in tournaments

Mycroft

Naia

2018

Random Struct Algorithms

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An oriented tree T on n vertices is unavoidable if every tournament on n vertices contains a copy of T. In this paper, we give a sufficient condition for T to be unavoidable, and use this to prove that almost all labeled oriented trees are unavoidable, verifying a conjecture of Bender and Wormald. We additionally prove that every tournament on n+o(n) vertices contains a copy of every oriented tree T on n vertices with polylogarithmic maximum degree, improving a result of Kühn, Mycroft, and Osthus.

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“…By Chebyshev's inequality, (12) and (14) it follows that P X − E(X) > ε 2 · E(X) ≤ Var(X) εE(X)/2 2 = o(1), which together with (12) proves the lemma.…”

Section: Most Oriented Trees Are Nicementioning

confidence: 62%

“…Indeed, Saks and Sós conjectured that every claw on n vertices with maximum degree at most

n / 2

is unavoidable. Lu gave a counterexample to this conjecture, but in the other direction showed that every claw with maximum degree at most

3 n / 8

is unavoidable. Lu, Wang, and Wong then extended these results by showing that every claw with maximum degree at most

19 n / 50

is unavoidable, but that there exist claws with maximum degree approaching

11 n / 23

which are avoidable.…”

Section: Introductionmentioning

confidence: 99%

Unavoidable trees in tournaments

Mycroft

Naia

2018

Random Struct Algorithms

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“…The answer is no. Indeed, it was conjectured in [13] that every tournament contains a spanning claw of depth at most 2, i.e., a spanning 1-tree of depth at most 2, and this was disproved in [7]. However, as a corollary of Theorem 1, we have the following result.…”

Section: A Short Surveymentioning

confidence: 68%

“…A simple example of an nunavoidable graph is a directed path of order n. This is just a restatement of a result of Re Âdei [11] that every tournament has a Hamiltonian path. Further examples of unavoidable graphs can be found in ( [1], [3], [4], [6], [7], [12], [13]). …”

Section: A Short Surveymentioning

confidence: 99%

Rooted Spanning Trees in Tournaments

Wang

Pan

et al. 2000

Graphs Comb

Self Cite

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A tournament of order n is an orientation of a complete graph with n vertices, and is speci®ed by its vertex set V T and edge set ET . A rooted tree is a directed tree such that every vertex except the root has in-degree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature. This result was strengthened to the following one: Every tournament contains a rooted spanning 2-tree of depth at most 2. We prove that every tournament of order n 800 contains a spanning rooted special 2-tree in this paper, where a rooted special 2-tree is a rooted 2-tree of depth 2 such that all except possibly one, non-root, non-leaf vertices, have out-degree 2 in the tree. A Short SurveyA tournament of order n is an orientation of a complete graph with n vertices, and is speci®ed by its vertex set V T and edge set ET . For an edge ab e ET , we also write a 3 b. When T is clear, we will simply write V and E for short. For A t V and B t V , we set AY B T fab e E j a e AY b e Bg.A rooted tree is a directed tree such that every vertex except the root has indegree 1, while the root has in-degree 0. A rooted k-tree is a rooted tree such that every vertex except the root has out-degree at most k; the out-degree of the root can be larger than k. A rooted 1-tree is also called a claw. The degree of a rooted tree is the out-degree of the root. For example, a claw of degree 1 is just a directed path. The depth of a rooted tree is the length of a longest directed path starting from the root.It is well-known that every tournament contains a rooted spanning tree of depth at most 2; and the root of such a tree is also called a king in the literature.

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“…Una trayectoria es unárbol dirigido donde para todo vértice v se tiene que Sea D una digráfica y P una trayectoria en D. Diremos que P es hamiltoniana si contiene a todos los vértices de D. 10 Figura 1.7. Trayectoria dirigida de orden 10, (1,6,7,2,3,8,9,4,5,10).…”

Section: Preliminaresunclassified

Árboles inevitables en torneos

Nol

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On claws belonging to every tournament

Cited by 11 publications

References 4 publications

Unavoidable trees in tournaments

Unavoidable trees in tournaments

Rooted Spanning Trees in Tournaments

Árboles inevitables en torneos

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