A study of $4-$cycle systems

TL;DR

This paper investigates the structure and connectivity of 4-cycle systems in complete graphs, focusing on trades of small volume and providing a matrix representation related to trade-vectors.

Contribution

It introduces the concept of 4-cycle trades of volume 2 and 3, proves the connectivity of all 4-cycle systems of order 9 via these trades, and presents a matrix with a null-space containing trade-vectors.

Findings

The set of all 4-cycle systems of order 9 is connected through trades of volume 2 and 3.

A full rank matrix with a null-space containing trade-vectors is constructed.

The study advances understanding of the structure and transformations of 4-cycle systems.

Abstract

A $4-$cycle system is a partition of the edges of the complete graph $K_n$ into $4-$cycles. Let ${ C}$ be a collection of cycles of length 4 whose edges partition the edges of $K_n$. A set of 4-cycles $T_1 \subset C$ is called a 4-cycle trade if there exists a set $T_2$ of edge-disjoint 4-cycles on the same vertices, such that $({C} \setminus T_1)\cup T_2$ also is a collection of cycles of length 4 whose edges partition the edges of $K_n$. We study $4-$cycle trades of volume two (double-diamonds) and three and show that the set of all 4-CS(9) is connected with respect of trading with trades of volume 2 (double-diamond) and 3. In addition, we present a full rank matrix whose null-space is containing trade-vectors.