A counterexample to a conjecture on edge-coloured tournaments

Galeana‐Sánchez, Hortensia; Rojas-Monroy, Rocío

doi:10.1016/j.disc.2003.11.015

Cited by 25 publications

(13 citation statements)

References 2 publications

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“…An example of a 4-colored tournament without rainbow triangles and without a monochromatic sink was given by GaleanaSánchez and Rojas-Monroy [2]. The question remains open when k = 3.…”

Section: Introductionmentioning

confidence: 98%

Monochromatic Sinks in k-Arc Colored Tournaments

Bland

2015

Graphs and Combinatorics

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Let T be a tournament on n vertices whose arcs are colored with k colors. A 3-cycle whose arcs are colored with three distinct colors is called a rainbow triangle. A rainbow triangle dominated by any nonempty set of vertices is called a dominated rainbow triangle. We prove that when n ≥ 5, if T does not contain a dominated rainbow triangle and all 4-and 5-cycles of T are near-monochromatic, then T has a monochromatic sink. We also prove that when n ≥ 4, if T does not contain a dominated rainbow triangle and all 4-cycles are monochromatic, then T has a monochromatic sink. A semi-cycle is a digraph C that either is a cycle or contains an arc x y such that C − x y + yx is a cycle. We prove that if n ≥ 4 and all 4-semi-cycles of T are near-monochromatic, then T has a monochromatic sink. We also show if n ≥ 5 and all 5-semi-cycles of T are near-monochromatic, then T has a monochromatic sink.

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“…An example of a 4-colored tournament without rainbow triangles and without a monochromatic sink was given by GaleanaSánchez and Rojas-Monroy [2]. The question remains open when k = 3.…”

Section: Introductionmentioning

confidence: 98%

Monochromatic Sinks in k-Arc Colored Tournaments

Bland

2015

Graphs and Combinatorics

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“…Afterwards, Sands et al [19] proposed the notion of kernels by monochromatic paths in 1982. For an arc-colored digraph D, define its kernel by monochromatic paths (or MP-kernel for short) to be a set S of vertices such that (i) no two vertices of S are connected by a monochromatic path in D, and (ii) every vertex outside S can reach S by a monochromatic path in D. It is worth noting that fruitful results on MP-kernels have been obtained in the past decades, we refer the reader to [11,12,14,15,19,20].…”

Section: Introductionmentioning

confidence: 99%

Kernels by rainbow paths in arc-colored tournaments

Bai

Zhang

2020

Discrete Applied Mathematics

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For an arc-colored digraph D, define its kernel by rainbow paths to be a set S of vertices such that (i) no two vertices of S are connected by a rainbow path in D, and (ii) every vertex outside S can reach S by a rainbow path in D. In this paper, we show that it is NP-complete to decide whether an arc-colored tournament has a kernel by rainbow paths, where a tournament is an orientation of a complete graph. In addition, we show that every arc-colored n-vertex tournament with all its strongly connected k-vertex subtournaments, 3 ≤ k ≤ n, colored with at least k − 1 colors has a kernel by rainbow paths, and the number of colors required cannot be reduced.

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“…The study of 1-colored kernels in m-colored digraphs already has a relatively extense literature and has explored sufficient conditions for the existence of such kernels in many infinite families of special digraphs as tournaments (particularly, in connection with so called Erdős' problem, see for instance [15], [14], [11] and [12]) and its generalizations (multipartite tournaments and quasi-transitive digraphs). As well, it has been of interest searching coloring conditions on subdigraphs of general digraphs to guarantee the existence of 1-colored kernels.…”

Section: Introductionmentioning

confidence: 99%

-colored kernels

Galena-SáNchez

Llano

Montellano-Ballesteros

2012

Discrete Applied Mathematics

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We study k-colored kernels in m-colored digraphs. An m-colored digraph D has k-colored kernel if there exists a subset K of its vertices such that (i) from every vertex v / ∈ K there exists an at most k-colored directed path from v to a vertex of K and (ii) for every u, v ∈ K there does not exist an at most k-colored directed path between them.In this paper, we prove that for every integer k ≥ 2 there exists a (k + 1)-colored digraph D without k-colored kernel and if every directed cycle of an m-colored digraph is monochromatic, then it has a k-colored kernel for every positive integer k. We obtain the following results for some generalizations of tournaments:(i) m-colored quasi-transitive and 3-quasi-transitive digraphs have a k-colored kernel for every k ≥ 3 and k ≥ 4, respectively (we conjecture that every m-colored l-quasi-transitive digraph has a k-colored kernel for every k ≥ l + 1), and (ii) m-colored locally in-tournament (out-tournament, respectively) digraphs have a k-colored kernel provided that every arc belongs to a directed cycle and every directed cycle is at most k-colored.

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A counterexample to a conjecture on edge-coloured tournaments

Cited by 25 publications

References 2 publications

Monochromatic Sinks in k-Arc Colored Tournaments

Monochromatic Sinks in k-Arc Colored Tournaments

Kernels by rainbow paths in arc-colored tournaments

-colored kernels

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