Paths through equally spaced points on a circle

TL;DR

This paper investigates the combinatorial properties of paths through equally spaced points on a circle, extending known conjectures and enumerations for path multisets and lengths up to 37 points, with special cases for prime and power-of-two sizes.

Contribution

It extends the verification of Buratti's and Horak-Rosa's conjectures to n ≤ 37 and advances understanding of the number of distinct path lengths for these configurations.

Findings

Confirmed conjectures for n ≤ 37.

Extended enumeration of path length possibilities up to n=37.

Proved uniqueness of path length for certain n when n is prime, twice a prime, or a power of 2.

Abstract

Consider $n$ points evenly spaced on a circle, and a path of $n-1$ chords that uses each point once. There are $m=\lfloor n/2\rfloor$ possible chord lengths, so the path defines a multiset of $n-1$ elements drawn from $\{1,2,\ldots,m\}$. The first problem we consider is to characterize the multisets which are realized by some path. Buratti conjectured that all multisets can be realized when $n$ is prime, and a generalized conjecture for all $n$ was proposed by Horak and Rosa. Previously the conjecture was proved for $n \leq 19$ and $n=23$; we extend this to $n\leq 37$ (OEIS sequence A352568). The second problem is to determine the number of distinct (euclidean) path lengths that can be realized. For this there is no conjecture; we extend current knowledge from $n\leq 16$ to $n\leq 37$ (OEIS sequence A030077). When $n$ is prime, twice a prime, or a power of 2, we prove that two paths have the same length only if they have the same multiset of chord lengths.