## Abstract

A [k, n, 1]-graph is a k-partite graph with parts of order n such that the bipartite graph induced by any pair of parts is a matching. An independent transversal in such a graph is an independent set that intersects each part in a single vertex. A factor of independent transversals is a set of n pairwise-disjoint independent transversals. Let f(k) be the smallest integer n_{0} such that every [k, n, 1]-graph has a factor of independent transversals assuming n ≥ n_{0} . Several known conjectures imply that for k ≥ 2, f(k) = k if k is even and f(k) = k + 1 if k is odd. While a simple greedy algorithm based on iterating Hall’s Theorem shows that f(k) ≤ 2k − 2, no better bound is known and in fact, there are instances showing that the bound 2k − 2 is tight for the greedy algorithm. Here we significantly improve upon the greedy algorithm bound and prove that f(k) ≤ 1.78k for all k sufficiently large, answering a question of MacKeigan.

Original language | American English |
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Article number | P4.23 |

Journal | Electronic Journal of Combinatorics |

Volume | 28 |

Issue number | 4 |

DOIs | |

State | Published - 2021 |

## All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Geometry and Topology
- Discrete Mathematics and Combinatorics
- Computational Theory and Mathematics
- Applied Mathematics