Cyclic subsets of tournaments

TL;DR

This paper investigates the probability that a randomly chosen induced subtournament of a high minimum degree tournament is Hamiltonian, providing optimal bounds and extending results to biased sampling measures.

Contribution

It establishes an optimal probability bound for Hamiltonian induced subtournaments in high-degree tournaments and extends the analysis to biased sampling regimes.

Findings

Derived an optimal probability bound for Hamiltonian subtournaments.

Extended results to non-uniform, p-biased sampling measures.

Generalized previous graph results to tournament settings.

Abstract

Let $G$ be a Dirac graph, and let $S$ be a vertex subset of $G$, chosen uniformly at random. How likely is the induced subgraph $G[S]$ to be Hamiltonian? This question, proposed by Erd\H{o}s and Faudree in 1996, was recently resolved by Dragani\'c, Keevash and M\"uyesser, in the setting of graphs. In this paper, we study a similar question for tournaments -- if $T$ is a tournament of high minimum degree, how likely is it for a random induced subtournament of $T$ to be Hamiltonian? We prove an optimal bound on this probability, and extend the results to the regime where the subset is not sampled uniformly at random, but according to a $p$-biased measure.