Powers of paths and cycles in tournaments

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 ε > 0 and every integer k, any tournament on n ≥ ε −Ck vertices which is ε-far from being transitive contains the k-th power of a cycle of length Ω(εn); both bounds are tight up to the implied constants.