## Abstract

Let {E_{1},…,E_{m}} be a partition of E(K_{n,n}), where K_{n,n} is the complete bipartite graph, and assume that [Formula presented]. It was conjectured in [1], that there exists a perfect matching M in K_{n,n} with [Formula presented] In this paper, we reprove combinatorially that this conjecture is true when m=2 or m=3. This result is proved in [1] by using topological methods. In the case m=4, we prove that there is always a perfect matching M in K_{n,n} with s(M)≤11. We also bring here an unpublished result from 2014 of the second author of this paper together with Irine Lo and Paul Seymour, proving that there exists a function of m alone, f(m), and a perfect matching M in K_{n,n} such that s(M)≤f(m). This result was later reproved by Alon in [2], where an explicit formulation of f(m) was given.

Original language | English |
---|---|

Article number | 113865 |

Journal | Discrete Mathematics |

Volume | 347 |

Issue number | 4 |

DOIs | |

State | Published - Apr 2024 |

### Bibliographical note

Publisher Copyright:© 2023 Elsevier B.V.

## Keywords

- Graphs matrices
- Latin-squares
- Perfect-matchings

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics