Powers of paths and cycles in tournaments

António Girão; Dániel Korándi; Alex Scott

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arXiv · 2105.12484

Powers of paths and cycles in tournaments

António Girão·Dániel Korándi·Alex Scott

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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 $Ω(\varepsilon n)$; both bounds are tight up to the implied constants.

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BibTeX RIS

António Girão, Dániel Korándi, Alex Scott. 2021-05-26. Powers of paths and cycles in tournaments. https://arxiv.org/abs/2105.12484

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