On Tournament Inversion

Abstract

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