### Abstract

Motivated by a scheduling problem that arises in the study of optical networks, we prove the following result, which is a variation of a conjecture of Haxell, Wilfong and Winkler. et k, n be two positive integers, let w _{sj}, 1 ≤ s ≤ n, 1 ≤ j ≤ k be nonnegative reals satisfying Σ_{j=1}^{k} w_{sj} < 1/n every 1 ≤ s ≤ n and let d_{sj} be arbitrary nonnegative reals. Then there are real numbers x_{1}, x_{2},...,x_{n} such that for every j, 1 ≤j ≤ k, the n cyclic closed intervals I_{s}^{(j)} = [x_{s} + d_{sj}, x_{s} + d_{dj} + w _{sj}], (1 ≤ s ≤ n), where the endpoints are reduced modulo 1, are pairwise disjoint on the unit circle. The proof is based on some properties of multivariate polynomials and on the validity of the Dyson conjecture.

Original language | English (US) |
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Pages (from-to) | 783-787 |

Number of pages | 5 |

Journal | Combinatorics Probability and Computing |

Volume | 15 |

Issue number | 5 |

DOIs | |

State | Published - Sep 1 2006 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Theoretical Computer Science
- Statistics and Probability
- Computational Theory and Mathematics
- Applied Mathematics