Classifying Intrinsically Linked Tournaments by Score Sequence

TL;DR

This paper investigates the classification of intrinsically linked tournaments based on their score sequences, identifying conditions under which tournaments are linkless and analyzing the distribution of such sequences.

Contribution

It introduces the concept of linkless score sequences and demonstrates that most 8-vertex tournaments have linkless score sequences, extending the analysis to larger tournaments.

Findings

Most 8-vertex tournament score sequences are linkless.

Existence of at least O(n) linkless score sequences for n vertices.

Conjecture that the fraction of linkless score sequences decreases as n grows.

Abstract

A tournament on 8 or more vertices may be intrinsically linked as a directed graph. We begin the classification of intrinsically linked tournaments by examining their score sequences. While many distinct tournaments may have the same score sequence, there exist score sequences $S$ such that any tournament with score sequence $S$ has an embedding with no nonsplit consistently oriented link. We call such score sequences $\textit{linkless}$, and we show that the vast majority of score sequences for 8 vertex tournaments are linkless. We also extend these results to $n$ vertex tournaments and are able to classify many longer score sequences as well. We show that for any $n$, there exist at least $O(n)$ linkless score sequences, but we conjecture that the fraction of score sequences of length $n$ that are linkless goes to 0 as $n$ becomes large.