Powers of paths in tournaments
Nemanja Dragani\'c, Fran\c{c}ois Dross, Jacob Fox, Ant\'onio Gir\~ao,, Fr\'ed\'eric Havet, D\'aniel Kor\'andi, William Lochet, David Munh\'a, Correia, Alex Scott, Benny Sudakov
TL;DR
This paper proves that every tournament contains a large k-th power of a directed path, improving previous bounds, and provides an exact solution for the case when k=2, establishing optimal path length bounds.
Contribution
It establishes that all tournaments contain the k-th power of a directed path of linear length and completely solves the case for k=2 with optimal bounds.
Findings
Every tournament contains the k-th power of a directed path of linear length.
For k=2, the path length is exactly eil(2n/3)-1, which is optimal.
Improves upon recent results by Yuster and Giraau.
Abstract
In this short note we prove that every tournament contains the -th power of a directed path of linear length. This improves upon recent results of Yuster and of Gir\~ao. We also give a complete solution for this problem when , showing that there is always a square of a directed path of length , which is best possible.
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