Quadrangularity and Strong Quadrangularity in Tournaments

TL;DR

This paper investigates the properties of quadrangularity in tournaments, establishing conditions for their existence and providing constructions for tournaments supporting orthogonal matrices.

Contribution

It characterizes when quadrangular and strongly quadrangular tournaments exist and introduces a construction method for such tournaments supporting orthogonal matrices.

Findings

Quadrangular tournaments exist for certain orders.

A necessary condition for supporting orthogonal matrices is identified.

Constructed examples of tournaments meeting the conditions.

Abstract

The pattern of a matrix M is a (0,1)-matrix which replaces all non-zero entries of M with a 1. A directed graph is said to support M if its adjacency matrix is the pattern of M. If M is an orthogonal matrix, then a digraph which supports M must satisfy a condition known as quadrangularity. We look at quadrangularity in tournaments and determine for which orders quadrangular tournaments exist. We also look at a more restrictive necessary condition for a digraph to support an orthogonal matrix, and give a construction for tournaments which meet this condition.