# Paths of given length in tournaments.

@article{Sah2020PathsOG, title={Paths of given length in tournaments.}, author={Ashwin Sah and Mehtaab Sawhney and Yufei Zhao}, journal={arXiv: Combinatorics}, year={2020} }

What is the maximum possible number of directed k-edge paths in an n-vertex tournament? We are interested in the regime of fixed k and large n. The expected number of directed k-edge paths in a uniform random n-vertex tournament is n(n− 1) · · · (n− k)/2k = (1+ o(1))n(n/2)k . In this short note we show that one cannot do better, thereby confirming an unpublished conjecture of Jacob Fox, Hao Huang, and Choongbum Lee. Theorem 1. Every n-vertex tournament has at most n(n/2)k directed k-edge paths.

## References

SHOWING 1-9 OF 9 REFERENCES

Tournaments with many Hamilton cycles

The object of interest is the maximum number, h(n), of Hamilton cycles in an n-tournament. By considering the expected number of Hamilton cycles in various classes of random tournaments, we obtain…

On the Number of Hamiltonian Cycles in a Tournament

- MathematicsCombinatorics, Probability and Computing
- 2005

Let $P(n)$ and $C(n)$ denote, respectively, the maximum possible numbers of Hamiltonian paths and Hamiltonian cycles in a tournament on n vertices. The study of $P(n)$ was suggested by Szele [14],…

Supersaturated graphs and hypergraphs

- Mathematics, Computer ScienceComb.
- 1983

We shall consider graphs (hypergraphs) without loops and multiple edges. Let ℒ be a family of so called prohibited graphs and ex (n, ℒ) denote the maximum number of edges (hyperedges) a graph…

On the maximum number of Hamiltonian paths in tournaments

- Computer ScienceRandom Struct. Algorithms
- 2001

By using the probabilistic method, we show that the maximum number of directed Hamiltonian paths in a complete directed graph with n vertices is at least (e− o(1)) n! 2n−1 .

Some advances on Sidorenko's conjecture

- Mathematics, Computer ScienceJ. Lond. Math. Soc.
- 2018

An embedding algorithm is developed which allows us to prove that bipartite graphs admitting a certain type of tree decomposition have Sidorenko’s property.

Sidorenko's conjecture for blow-ups

- Mathematics
- 2018

A celebrated conjecture of Sidorenko and Erdős-Simonovits states that, for all bipartite graphs H, quasirandom graphs contain asymptotically the minimum number of copies of H taken over all graphs…

A correlation inequality for bipartite graphs

- Mathematics, Computer ScienceGraphs Comb.
- 1993

This inequality states that the random graph with fixed numbers of vertices and edges contains the asymptotically minimal number of copies of G.

Kombinatorische Untersuchungen über den gerichteten vollständigen Graphen

- Mat. Fiz. Lapok
- 1943