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Published online by Cambridge University Press: 31 July 2006
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.
Let $k,n$ be two positive integers, let $w_{sj}, 1 \leq s \leq n, 1 \leq j \leq k$ be nonnegative reals satisfying $\sum_{j=1}^k w_{sj}< 1/n$ for every $1 \leq s \leq n$ and let $d_{sj}$ be arbitrary nonnegative reals. Then there are real numbers $x_1, x_2, {\ldots}\,,x_n$ such that for every $j$, $1 \leq j \leq k$, the $n$ cyclic closed intervals $I_s^{(j)}=[x_s+d_{sj},x_s+d_{sj}+w_{sj}]$, $(1 \leq s \leq n)$, 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.