2022/05/12 by Brendan D. McKay, T. J. Peters, McKay, Brendan D. +1 · 1 citation
Computer Science · Mathematics · #11Axx #11Y55 #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Limits and Structures in Graph Theory #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2205.06004
openalex publication_date 2022/05/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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,…,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 ≤ 19 and n=23; we extend this to n≤ 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≤ 16 to n≤ 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.