Dominant tournament families

TL;DR

This paper characterizes large classes of tournament families that are prevalent in large random tournaments, providing explicit constructions and identifying conditions under which families are dominant.

Contribution

It introduces the concept of dominant families of tournaments and characterizes several large such families for all sufficiently large h, including subgraph containment and feedback arc set size conditions.

Findings

Families containing large subgraphs are dominant for large h.

Families with bounded feedback arc set size are dominant.

Explicit constructions of dominant families for small h.

Abstract

For a tournament $H$ with $h$ vertices, its typical density is $h!2^{-\binom{h}{2}}/aut(H)$, i.e. this is the expected density of $H$ in a random tournament. A family ${\mathcal F}$ of $h$-vertex tournaments is {\em dominant} if for all sufficiently large $n$, there exists an $n$-vertex tournament $G$ such that the density of each element of ${\mathcal F}$ in $G$ is larger than its typical density by a constant factor. Characterizing all dominant families is challenging already for small $h$. Here we characterize several large dominant families for every $h$. In particular, we prove the following for all $h$ sufficiently large: (i) For all tournaments $H^*$ with at least $5\log h$ vertices, the family of all $h$-vertex tournaments that contain $H^*$ as a subgraph is dominant. (ii) The family of all $h$-vertex tournaments whose minimum feedback arc set size is at most $\frac{1}{2}\binom{h}{2}-h^{3/2}\sqrt{\ln h}$ is dominant. For small $h$, we construct a dominant family of $6$ (i.e. $50\%$ of the) tournaments on $5$ vertices and dominant families of size larger than $40\%$ for $h=6,7,8,9$. For all $h$, we provide an explicit construction of a dominant family which is conjectured to obtain an absolute constant fraction of the tournaments on $h$ vertices. Some additional intriguing open problems are presented.