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

Paths Through Equally Spaced Points on a Circle

Brendan D. McKay 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.

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(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.