An improvement of Lichiardopol’s theorem on disjoint cycles in tournaments

Ma, Fuhong; Jin, Yan

doi:10.1016/j.amc.2018.10.086

Cited by 3 publications

(4 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Problem For any given integers

k\ge 1

and

q\ge 3

, does any (regular) tournament with

{\delta }^{+}(T)\ge qk\unicode{x02215}2

contains

k

disjoint cycles of length

q

? Ma and Yan [14] proved that for large enough integers

q

and

k

, every tournament with both minimum indegree and outdegree more than

((q-1)k-1)\unicode{x02215}2

contains

k-q

disjoint

q

‐cycles.…”

Section: Open Problemsmentioning

confidence: 99%

See 1 more Smart Citation

Vertex‐disjoint cycles of the same length in tournaments

Wang

Jin

2022

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

A tournament is a digraph which has exactly one arc between any two vertices. Let k and q be two positive integers with ≥ q 4. A q-cycle is a cycle of length q. We prove that for any real number ∕ α q > 2, there exists a constant k α such that for every ≥ k k α , any tournament T with minimum outdegree at least αk contains k disjoint q-cycles. This generalizes a result by Bang-Jensen, Bessy, and Thomassé for disjoint 3-cycles. The linear factor ∕ q 2 of minimum outdegree is the best possible.to denote the minimum outdegree of D. A number of digraphs are said to be vertex-disjoint (briefly disjoint) if no two of them have any common vertex. Similarly, a number of digraphs are called arc-disjoint if they have no common arc.It is NP-complete to decide whether a digraph has k disjoint cycles for an integer k in [2]. It turns out that we have to seek sufficient conditions to guarantee the existence of disjoint cycles in a digraph. The following beautiful conjecture of Bermond and Thomassen states that a digraph with large minimum outdegree must have many disjoint cycles.Conjecture 1 (Bermond and Thomassen [5]). For a given integerObserve that the complete digraph of order k 2 − 1 (with all possible arcs) shows that this minimum outdegree condition is the best possible. The conjecture is trivial for k = 1. It was

show abstract

“…Problem For any given integers

k\ge 1

and

q\ge 3

, does any (regular) tournament with

{\delta }^{+}(T)\ge qk\unicode{x02215}2

contains

k

disjoint cycles of length

q

? Ma and Yan [14] proved that for large enough integers

q

and

k

, every tournament with both minimum indegree and outdegree more than

((q-1)k-1)\unicode{x02215}2

contains

k-q

disjoint

q

‐cycles.…”

Section: Open Problemsmentioning

confidence: 99%

“…For any given integers ≥ k 1 and ≥ q 3, does any (regular) tournament with ≥ ∕ δ T qk ( ) 2 + contains k disjoint cycles of length q? Ma and Yan [14] proved that for large enough integers q and k, every tournament with both minimum indegree and outdegree more than…”

mentioning

confidence: 99%

Vertex‐disjoint cycles of the same length in tournaments

Wang

Jin

2022

Journal of Graph Theory

Self Cite

View full text Add to dashboard Cite

show abstract

“…If q 3 and k 1, then every tournament with minimum outdegree and indegree both at least (q − 1)k − 1 contains k disjoint q-cycles. [20] improved Theorem 1.4 by guaranteeing more than k disjoint cycles under the same conditions, so the conclusion of Theorem 1.4 is not sharp ( [21] addressed the special case of 4-cycles in regular tournaments).…”

Section: Theorem 13 ([3]mentioning

confidence: 99%

Lichiardopol’s Conjecture on Disjoint Cycles in Tournaments

Ma¹,

West²,

Jin³

2020

Electron. J. Combin.

View full text Add to dashboard Cite

show abstract

“…In 2018, Bucić [8] enhanced the result of Alon by showing

f(k)\le 18k

, which is the best‐known result. Since then, several results on Conjecture 1.1 in terms of special families digraphs such as tournaments see, for example, [3, 7, 10, 11, 14, 15] and bipartite tournaments [2] have been obtained.…”

Section: Introductionmentioning

confidence: 99%

Disjoint cycles in tournaments and bipartite tournaments

Chen,

Chang

2023

Journal of Graph Theory

View full text Add to dashboard Cite

A conjecture proposed by Bermond and Thomassen in 1981 states that every digraph with minimum out‐degree at least contains vertex disjoint directed cycles for any integer , which has received substantial attention. This conjecture has been confirmed for . In 2014, Lichiardopol raised a very related conjecture that for every integer , there exists an integer such that every digraph with minimum out‐degree at least contains vertex disjoint directed cycles of different lengths. For a digraph and a set of vertex disjoint directed cycles in , we denote to be the maximum number of directed cycles in of distinct lengths. Let . We define if has no vertex disjoint directed cycles. In this paper, we mainly investigate vertex disjoint directed cycles in tournaments and bipartite tournaments. We first show that for every tournament with minimum out‐degree at least , where . We further prove that for and , any tournament with minimum out‐degree at least satisfies that . Moreover, we deduce that for any tournament with minimum out‐degree at least 7, holds. Additionally, we classify strong bipartite tournaments with minimum out‐degree at least in which any vertex disjoint directed cycles have the same length, where . That is, for any strong bipartite tournament with minimum out‐degree at least , then if and only if is isomorphic to a member of , which is defined in the context.

show abstract

An improvement of Lichiardopol’s theorem on disjoint cycles in tournaments

Cited by 3 publications

References 18 publications

Vertex‐disjoint cycles of the same length in tournaments

Vertex‐disjoint cycles of the same length in tournaments

Lichiardopol’s Conjecture on Disjoint Cycles in Tournaments

Disjoint cycles in tournaments and bipartite tournaments

Contact Info

Product

Resources

About