All Feedback Arc Sets of a Random Turán Tournament Have $\lfloor {n}/{k}\rfloor-{k}+1$ Disjoint ${k}$-Cliques (and This Is Tight)

Abstract

We look at 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 {\em all} feedback arc sets of $G$. In particular, to make $G$ acyclic, one must remove (or reverse) $f_H(G)$ pairwise disjoint copies of $H$. Most intriguing is the case where $H$ is a $k$-clique, where the parameter is denoted by $f_k(G)$. Determining $f_k(G)$ for arbitrary $G$ seems challenging. Here we determine $f_k(G)$ precisely for almost all $k$-partite tournaments. Let $s(G)$ denote the size of the smallest vertex class of a $k$-partite tournament $G$. We prove that for all sufficiently large $s=s(G)$, a random $k$-partite tournament $G$ satisfies $f_k(G) = s(G)-k+1$ almost surely. 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 Tur\'an graph $T(n,k)$.