Classifying Intrinsically Linked Tournaments by Score Sequence

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.