## Abstract

An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let k,ℓ be positive integers. For a tournament T, let f_{T}(k) be the largest integer such that every k-edge coloring of T has a path-monochromatic subtree with at least f_{T}(k) vertices and let f_{T}(k,ℓ) be the restriction to subtrees of depth at most ℓ. It was proved by Landau that f_{T}(1,2)=n and proved by Sands et al. that f_{T}(2)=n where |V(T)|=n. Here we consider f_{T}(k) and f_{T}(k,ℓ) in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-Häggkvist Conjecture. We also study the typical value of f_{T}(k) and f_{T}(k,ℓ), i.e., when T is a random tournament.

Original language | English |
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Article number | 114022 |

Journal | Discrete Mathematics |

Volume | 347 |

Issue number | 6 |

DOIs | |

State | Published - Jun 2024 |

### Bibliographical note

Publisher Copyright:© 2024 Elsevier B.V.

## Keywords

- Edge coloring
- Monochromatic path
- Tournament

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics