SIAM Journal on Discrete Mathematics, Volume 35, Issue 2, Page 1460-1477, January 2021.
What must one do in order to make acyclic a given oriented graph? Here we look at the structures that must be removed (or reversed) in order to make acyclic a given oriented graph. For a directed acyclic graph $H$ and an oriented graph $G$, let $f_H(G)$ be the maximum number of pairwise disjoint copies of $H$ that can be found in all feedback arc sets of $G$. In particular, to make $G$ acyclic, one must at least remove (or reverse) $f_H(G)$ pairwise disjoint copies of $H$. Perhaps most intriguing is the case where $H$ is a $k$-clique, in which case the parameter is denoted by $f_k(G)$. Determining $f_k(G)$ for arbitrary $G$ seems challenging. Here we essentially answer the problem, precisely, for the family of $k$-partite tournaments. Let $s(G)$ denote the size of the smallest vertex class of a $k$-partite tournament $G$. It is not difficult to show that $f_k(G) \le s(G)-k+1$ (assume that $s(G) \ge k-1$). Our main result is that for all sufficiently large $s=s(G)$, there are $k$-partite tournaments for which $f_k(G) = s(G)-k+1$. In fact, much more can be said: A random $k$-partite tournament $G$ satisfies $f_k(G) = s(G)-k+1$ almost surely (i.e., with probability tending to $1$ as $s(G)$ goes to infinity). In particular, as the title states, $f_k(G) = \lfloor n/k\rfloor-k+1$ almost surely, where $G$ is a random orientation of the Turán graph $T(n,k)$.

中文翻译：

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随机 Turán 锦标赛的所有反馈弧集都有 $\lfloor {n}/{k}\rfloor-{k}+1$ 不相交的 ${k}$-Cliques（这是紧密的）

SIAM 离散数学杂志，第 35 卷，第 2 期，第 1460-1477 页，2021 年 1 月。
为了使给定的有向图无环，必须做什么？在这里，我们看看必须删除（或反转）的结构，以便使给定的有向图无环。对于有向无环图 $H$ 和有向图 $G$，令 $f_H(G)$ 是 $G$ 的所有反馈弧集中可以找到的 $H$ 的成对不相交副本的最大数量。特别是，要使 $G$ 无环，必须至少删除（或反转）$f_H(G)$ 成对不相交的 $H$ 副本。也许最有趣的是 $H$ 是 $k$-clique 的情况，在这种情况下，参数由 $f_k(G)$ 表示。为任意 $G$ 确定 $f_k(G)$ 似乎具有挑战性。在这里，我们基本上准确地为 $k$-partite 锦标赛系列回答了这个问题。让 $s(G)$ 表示 $k$-partite 锦标赛 $G$ 的最小顶点类的大小。不难证明$f_k(G) \le s(G)-k+1$（假设$s(G) \ge k-1$）。我们的主要结果是，对于所有足够大的 $s=s(G)$，存在 $k$-partite 锦标赛，其中 $f_k(G) = s(G)-k+1$。事实上，可以说的更多：随机 $k$-partite 锦标赛 $G$ 几乎肯定满足 $f_k(G) = s(G)-k+1$（即，概率趋向于 $1$ 作为 $s (G)$ 趋于无穷）。特别是，正如标题所述，$f_k(G) = \lfloor n/k\rfloor-k+1$ 几乎可以肯定，其中 $G$ 是图兰图 $T(n,k)$ 的随机方向。随着 $s(G)$ 趋于无穷大，概率趋于 $1$）。特别是，正如标题所述，$f_k(G) = \lfloor n/k\rfloor-k+1$ 几乎可以肯定，其中 $G$ 是图兰图 $T(n,k)$ 的随机方向。随着 $s(G)$ 趋于无穷大，概率趋于 $1$）。特别是，正如标题所述，$f_k(G) = \lfloor n/k\rfloor-k+1$ 几乎可以肯定，其中 $G$ 是图兰图 $T(n,k)$ 的随机方向。