On tournament inversion

Abstract

An {\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\rm inv}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$ that result in the transitive tournament. Let ${\rm inv}_k(n)$ be the maximum of ${\rm inv}_k(T)$ taken over $n$-vertex tournaments. It is well-known that ${\rm inv}_2(n)=(1+o(1))n^2/4$ and it was recently proved by Alon et al. that ${\rm inv}(n):={\rm inv}_{n}(n)=n(1+o(1))$. In these two extreme cases ($k=2$ and $k=n$), random tournaments are asymptotically extremal objects. It is proved that the random tournament {\em does not} asymptotically attain ${\rm inv}_k(n)$ when $k \ge k_0$ and conjectured that ${\rm inv}_3(n)$ is (only) attained by (quasi) random tournaments. It is further proved that $(1+o(1)){\rm inv}_3(n)/n^2 \in [\frac{1}{12}, 0.0992)$ and $(1+o(1)){\rm inv}_k(n)/n^2 \in [\frac{1}{2k(k-1)}+\delta_k, \frac{1}{2 \lfloor k^2/2 \rfloor}-\epsilon_k]$ where $\epsilon_k > 0$ for all $k \ge 3$ and $\delta_k > 0$ for all $k \ge k_0$.