Upper bounds for the necklace folding problems

TL;DR

This paper disproves a conjecture about the maximum ratio of beads covered in necklace foldings, providing a counterexample that shows the ratio is at most approximately 0.5858, with implications for related matching problems.

Contribution

It presents the first counterexample to the conjecture that the ratio is 2/3, establishing a new upper bound for the necklace folding problem.

Findings

Counterexample shows ratio ≤ 2 - √2 ≈ 0.5858

Applicable to both separated and homogeneous models

Extends to cases with unequal color class sizes

Abstract

A necklace can be considered as a cyclic list of $n$ red and $n$ blue beads in an arbitrary order, and the goal is to fold it into two and find a large cross-free matching of pairs of beads of different colors. We give a counterexample for a conjecture about the necklace folding problem, also known as the separated matching problem. The conjecture (given independently by three sets of authors) states that $\mu=\frac{2}{3}$, where $\mu$ is the ratio of the `covered' beads to the total number of beads. We refute this conjecture by giving a construction which proves that $\mu \le 2 \nolinebreak - \nolinebreak \sqrt 2 < 0.5858$. Our construction also applies to the homogeneous model: when we are matching beads of the same color. Moreover, we also consider the problem where the two color classes not necessarily have the same size.