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On tournament inversion
Raphael Yuster
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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 invₖ(T) be the minimum length of a sequence of inversions using sets of size at most k that result in the transitive tournament. Let invₖ(n) be the maximum of invₖ(T) taken over n-vertex tournaments. It is well-known that inv₂(n)=(1+o(1))n²/4 and it was recently proved by Alon et al. that inv(n):=invₙ(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 invₖ(n) when k ≥k₀ and conjectured that inv₃(n) is (only) attained by (quasi) random tournaments. It is further proved that (1+o(1))inv₃(n)/n² ∈[1/12, 0.0992) and (1+o(1))invₖ(n)/n² ∈[1/(2k(k-1))+δₖ, 1/(2 ⌊k²/2 ⌋)-ϵₖ] where ϵₖ > 0 for all k ≥3 and δₖ > 0 for all k ≥k₀.
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