论文标题
路径穿过圆上同样间隔的点
Paths through equally spaced points on a circle
论文作者
论文摘要
考虑$ n $点在一个圆上均匀间隔，以及一次使用每个点的$ N-1 $和弦的路径。有$ m = \ lfloor n/2 \ rfloor $可能的和弦长度，因此该路径定义了从$ \ {1,2，\ ldots，m \} $绘制的$ n-1 $元素的多序列。我们考虑的第一个问题是表征多个路径实现的多集。 Buratti猜想，当$ n $是Prime时，所有多组都可以实现，并且Horak和Rosa提出了所有$ N $的广义猜想。以前，猜想被证明为$ n \ leq 19 $和$ n = 23 $;我们将其扩展到$ n \ leq 37 $（OEIS序列A352568）。第二个问题是确定可以实现的不同（欧几里得）路径长度的数量。为此，没有猜想。我们将当前知识从$ n \ leq 16 $扩展到$ n \ leq 37 $（OEIS序列A030077）。当$ n $是素数，两倍的素数或2个功率时，我们证明，只有在它们具有相同的和弦长度时，两条路径的长度相同。
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.
