AI 中文总结
本文解决了关于路径反转竞赛图逆序数的猜想，该竞赛图由自然传递竞赛图反转特定连续对得到，猜想自然路径反转族的逆序数为\(\left\lfloor(n - 1)/2\right\rfloor\)，通过相关研究得出了结论。
AI 中文摘要
设\(D\)为一个竞赛图，\(X\subseteq V(D)\)。\(X\)的反转操作会反转两端点都在\(X\)中的所有弧，其他弧不变。若一系列反转操作能产生一个无环（即传递的）竞赛图，则称这一系列反转操作是一个去环族。逆序数\(\inv(D)\)是这样一个族的最小规模。设\(Q_n\)是在\([n]\)上通过恰好反转连续对\(12,23,\ldots,(n - 1)n\)从自然传递竞赛图得到的竞赛图。贝尔克希内、布阿齐兹、布达布斯和普泽在未发表的手稿中猜想一个自然路径反转族的逆序数恰好为\(\left\lfloor(n - 1)/2\right\rfloor\)。本文解决了这个猜想。
英文摘要
Let $D$ be a tournament and let $X\subseteq V(D)$. The inversion of $X$ reverses all arcs whose both endpoints lie in $X$ and leaves every other arc unchanged. A family of inversions is a decycling family if applying all of them produces an acyclic, equivalently transitive, tournament. The inversion number $\inv(D)$ is the minimum size of such a family. Let $Q_n$ be the tournament on $[n]$ obtained from the natural transitive tournament by reversing precisely the consecutive pairs $12,23,\ldots,(n-1)n$. Belkhechine, Bouaziz, Boudabbous, and Pouzet conjectured in their unpublished manuscript that a natural path-reversed family has inversion number exactly $\left\lfloor(n-1)/2\right\rfloor$. The same problem was later recorded by Bang-Jensen, da Silva, and Havet and by Alon, Powierski, Savery, Scott, and Wilmer. In this paper we resolve this conjecture.
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