Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Vertex-disjoint cycles of different lengths in tournaments

Yandong Bai, Wenpei Jia

Abstract

Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles,here $k$ is a positive integer. Lichiardopol conjectured in 2014 that for every positive integer $k$ there exists an integer $g(k)$ such that every digraph with minimum outdegree at least $g(k)$ contains $k$ vertex-disjoint cycles of different lengths. Recently, Chen and Chang proved in [J. Graph Theory 105 (2) (2024) 297-314] that for $k\geqslant 3$ every tournament with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles in which two of them have different lengths. Motivated by the above two conjectures and related results, we investigate vertex-disjoint cycles of different lengths in tournaments, and show that when $k\geqslant 5$ every tournament with minimum outdegree at least $2k-1$ contains $k$ vertex-disjoint cycles in which three of them have different lengths. In addition, we show that every tournament with minimum outdegree at least $6$ contains three vertex-disjoint cycles of different lengths and the minimum outdegree condition is sharp. This answers a question proposed by Chen and Chang.

Vertex-disjoint cycles of different lengths in tournaments

Abstract

Bermond and Thomassen conjectured in 1981 that every digraph with minimum outdegree at least contains vertex-disjoint cycles,here is a positive integer. Lichiardopol conjectured in 2014 that for every positive integer there exists an integer such that every digraph with minimum outdegree at least contains vertex-disjoint cycles of different lengths. Recently, Chen and Chang proved in [J. Graph Theory 105 (2) (2024) 297-314] that for every tournament with minimum outdegree at least contains vertex-disjoint cycles in which two of them have different lengths. Motivated by the above two conjectures and related results, we investigate vertex-disjoint cycles of different lengths in tournaments, and show that when every tournament with minimum outdegree at least contains vertex-disjoint cycles in which three of them have different lengths. In addition, we show that every tournament with minimum outdegree at least contains three vertex-disjoint cycles of different lengths and the minimum outdegree condition is sharp. This answers a question proposed by Chen and Chang.
Paper Structure (5 sections, 4 theorems, 40 equations)

This paper contains 5 sections, 4 theorems, 40 equations.

Table of Contents

  1. Introduction
  2. Additional terminology and notation
  3. Proof of Theorem \ref{['thm:main']}
  4. Proof of Theorem \ref{['thm:main2']}
  5. Concluding remarks

Key Result

Theorem 1

Every tournament with minimum outdegree at least $2k-1$ contains $k$ disjoint cycles in which three of them have different lengths, here $k\geqslant 5$ is an integer.

Theorems & Definitions (28)

  • Conjecture 1: Bermond and Thomassen BT
  • Conjecture 2: Lichiardopol L
  • Theorem 1
  • Theorem 2
  • Lemma 1: Moon M
  • Lemma 2
  • proof
  • Claim 1
  • proof
  • Claim 2
  • ...and 18 more