An improved bound for disjoint directed cycles

Bucić, Matija

doi:10.1016/j.disc.2018.04.027

Cited by 15 publications

(8 citation statements)

References 7 publications

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“…Alon [1] gave a linear bound by proving that 64k suffices (Lemma 11). Recently, Bucić [6] proved a better bound 18k towards this conjecture. One may find that if we apply Bucić's new bound instead of Alon's bound to our proof of Theorem 8, then we can improve the constant in the second term of Theorem 8.…”

Section: Suppose That the Bipartition Ofmentioning

confidence: 95%

On sufficient conditions for rainbow cycles in edge-colored graphs

Fujita

Ning

et al. 2019

Discrete Mathematics

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Let G be an edge-colored graph. We use e(G) and c(G) to denote the number of edges of G and the number of colors appearing on E(G), respectively. For a vertex v ∈ V (G), the color neighborhood of v is defined as the set of colors assigned to the edges incident to v. A subgraph of G is rainbow if all of its edges are assigned with distinct colors. The well-known Mantel's theorem states that a graph G on n vertices contains a triangle if e(G) ≥ ⌊ n 2 4 ⌋ + 1. Rademacher (1941) showed that G contains at least ⌊ n 2 ⌋ triangles under the same condition. Li, Ning, Xu and Zhang (2014) proved a rainbow version of Mantel's theorem: An edge-colored graph G has a rainbow triangle if e(G) + c(G) ≥ n(n + 1)/2. In this paper, we first characterize all graphs G satisfying e(G) + c(G) ≥ n(n + 1)/2 − 1 but containing no rainbow triangles.Motivated by Rademacher's theorem, we then characterize all graphs G which satisfy e(G) + c(G) ≥ n(n + 1)/2 but contain only one rainbow triangle. We further obtain two results on color neighborhood conditions for the existence of rainbow short cycles.Our results improve a previous theorem due to Broersma, Li, Woeginger, and Zhang (2005). Moreover, we provide a sufficient condition in terms of color neighborhood for the existence of a specified number of vertex-disjoint rainbow cycles.

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Section: Suppose That the Bipartition Ofmentioning

confidence: 95%

On sufficient conditions for rainbow cycles in edge-colored graphs

Fujita

Ning

et al. 2019

Discrete Mathematics

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“…, 1) k 64k. Recently, this bound has been improved to 24k by Bucić [5]. In view of Proposition 5.3, we believe that considering (a, 2)-feasible partitions in edgecoloured graphs could be a reasonable approach for improving Bucić's result concerning Bermond and Thomassen's conjecture in digraphs.…”

Section: Claim 9 There Exists a Vertexmentioning

confidence: 82%

“…Bermond and Thomassen [4] conjectured that f (k) = 2k − 1, Alon [1] showed that f (k) 64k and recently Bucić [5] improved this bound to 24k.…”

Section: Theorem 13 (Thomassen [14]mentioning

confidence: 99%

Decomposing edge-coloured graphs under colour degree constraints

Fujita

Wang

2019

Combinator. Probab. Comp.

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For an edge-coloured graph G, the minimum colour degree of G means the minimum number of colours on edges which are incident to each vertex of G. We prove that if G is an edge-coloured graph with minimum colour degree at least 5, then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum colour degree at least 2. We show this theorem by proving amuch stronger form. Moreover, we point out an important relationship between our theorem and Bermond and Thomassen’s conjecture in digraphs.

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“…It was proved for

k=2

by Thomassen [16] and for

k=3

by Lichiardopol, Pór, and Sereni [13]. On the other hand, Alon [1] showed that the lower bound

64k

on the minimum outdegree suffices to give

k

disjoint cycles and later Bucić [6] reduced this to

18k

.…”

Section: Introductionmentioning

confidence: 99%

“…proved for k = 2 by Thomassen [16] and for k = 3 by Lichiardopol, Pór, and Sereni [13]. On the other hand, Alon [1] showed that the lower bound k 64 on the minimum outdegree suffices to give k disjoint cycles and later Bucić [6] reduced this to k 18 . Recently, the conjecture was verified for tournaments by Bang-Jensen, Bessy, and Thomassé [3], here a tournament is a digraph which has exactly one arc between any two vertices.…”

mentioning

confidence: 99%

Vertex‐disjoint cycles of the same length in tournaments

Wang

Jin

2022

Journal of Graph Theory

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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

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An improved bound for disjoint directed cycles

Cited by 15 publications

References 7 publications

On sufficient conditions for rainbow cycles in edge-colored graphs

On sufficient conditions for rainbow cycles in edge-colored graphs

Decomposing edge-coloured graphs under colour degree constraints

Vertex‐disjoint cycles of the same length in tournaments

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