## Abstract

Let script G sign(r, n) denote the set of all r-partite graphs consisting of n vertices in each partite class. An independent transversal of G ∈ script G sign(r, n) is an independent set consisting of exactly one vertex from each vertex class. Let Δ(r, n) be the maximal integer such that every G ∈ script G sign(r, n) with maximal degree less than Δ(r, n) contains an independent transversal. Let C_{r} = lim_{n→∞} Δ(r, n)/n. We establish the following upper and lower bounds on C_{r}, provided r > 2: 2^{[log r]-1}/2[log r] - 1 ≥ C_{r} ≥ max {1/2e, 1/2[log(r/3)], 1/3 . 2[log r]-3}. For all r > 3, both upper and lower bounds improve upon previously known bounds of Bollobás, Erdos and Szemerédi. In particular, we obtain that C_{4} = 2/3, and that lim_{r→∞} C_{r} ≥ 1/(2e), where the last bound is a consequence of a lemma of Alon and Spencer. This solves two open problems of Bollobás, Erdos and Szemerédi.

Original language | English |
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Pages (from-to) | 255-261 |

Number of pages | 7 |

Journal | Discrete Mathematics |

Volume | 176 |

Issue number | 1-3 |

DOIs | |

State | Published - 15 Nov 1997 |

Externally published | Yes |

## ASJC Scopus subject areas

- Theoretical Computer Science
- Discrete Mathematics and Combinatorics