Various Theorems on Tournaments

TL;DR

This thesis explores structural properties of prime and matching tournaments, proving the existence of certain subtournaments, connectivity of cyclic triangles, and characterizations of tournaments excluding specific configurations.

Contribution

It introduces new theorems on prime tournaments, cyclic triangle sequences, and conditions for matching tournaments, along with a structure theorem for tournaments excluding specific subgraphs.

Findings

Existence of prime subtournaments with increasing size

Connectivity of cyclic triangles in prime tournaments

Characterization of tournaments excluding K_n and K_n^*

Abstract

In this thesis we prove a variety of theorems on tournaments. A \emph{prime} tournament is a tournament $G$ such that there is no $X \subseteq V(G)$, $1 < |X| < |V(G)|$, such that for every vertex $v \in V(G) \minus X$, either $v \ra x$ for all $x \in X$ or $x \ra v$ for all $x \in X$. First, we prove that given a prime tournament $G$ which is not in one of three special families of tournaments, for any prime subtournament $H$ of $G$ with $5 \le |V(H)| < |V(G)|$ there exists a prime subtournament of $G$ with $|V(H)| + 1$ vertices that has a subtournament isomorphic to $H$. We next prove that for any two cyclic triangles $C$, $C^\prime$ in a prime tournament $G$, there is a sequence of cyclic triangles $C_1,...,C_n$ such that $C_1 = C$, $C_n = C^\prime$, and $C_i$ shares an edge with $C_{i+1}$ for all $1 \le i \le n-1$. Next, we consider what we call \emph{matching tournaments}, tournaments whose vertices can be ordered in a horizontal line so that every vertex is the head or tail of at most one edge that points right-to-left. We determine the conditions under which a tournament can have two different orderings satisfying the above conditions. We also prove that there are infinitely many minimal tournaments that are not matching tournaments. Finally, we consider the tournaments $K_n$ and $K_n^\ast$, which are obtained from the transitive tournament with $n$ vertices by reversing the edge from the second vertex to the last vertex and from the first vertex to the second-to-last vertex, respectively. We prove a structure theorem describing tournaments which exclude $K_n$ and $K_n^\ast$ as subtournaments.