Powers of paths and cycles in tournaments

TL;DR

This paper proves that any tournament can be partitioned into a bounded number of $k$-th powers of paths and that large, far-from-transitive tournaments contain long $k$-th power cycles, with bounds that are essentially tight.

Contribution

It establishes tight bounds on the partitioning of tournaments into $k$-th powers of paths and on the existence of long $k$-th power cycles in far-from-transitive tournaments.

Findings

Any tournament can be partitioned into at most $2^{ck}$ $k$-th powers of paths.

Tournaments far from transitive contain long $k$-th power cycles proportional to their size.

Bounds are tight up to constants.

Abstract

We show that for every positive integer $k$, any tournament can be partitioned into at most $2^{ck}$ $k$-th powers of paths. This result is tight up to the exponential constant. Moreover, we prove that for every $\varepsilon>0$ and every integer $k$, any tournament on $n\ge \varepsilon^{-Ck}$ vertices which is $\varepsilon$-far from being transitive contains the $k$-th power of a cycle of length $\Omega(\varepsilon n)$; both bounds are tight up to the implied constants.