Bisection of Circle Colorings

Abstract

Consider $2n$ beads of k colors arranged on a necklace, using $2a$, beads of color i. A bisection is a set of disjoint strings (“intervals”) of beads whose union captures half the beads of each color. We prove that any arrangement with k colors has a bisection using at most $\lceil k / 2 \rceil $ intervals. In addition, if k is odd, an endpoint of one interval can be specified arbitrarily. The result is best possible. For fixed k, there is a polynomial-time algorithm to find such a bisection; it runs in $O ( n^{k - 2} )$ for $k\geqq 3$. We consider continuous and linear versions of the problem and use them to obtain applications in geometry, VLSI circuit design, and orthogonal functions.