Journal of Integer Sequences, Vol. 25 (2022), Article 22.7.4 |

School of Computing

Australian National University

Canberra, ACT 2601

Australia

Tim Peters

Python Software Foundation

USA

**Abstract:**

Consider *n* points evenly spaced on a circle,
and a path of *n* – 1
chords that uses each point once.
There are *m* = ⌊*n*/2⌋ 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.

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.
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.

(Concerned with sequences A030077 A352568.)

Received May 12 2022; revised versions received September 13 2022; September 14 2022.
Published in *Journal of Integer Sequences*,
September 15 2022.

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