A linear algebraic approach to 4–cycle systems

A 4-cycle system is a partition of the edges of the complete graph Kn into 4-cycles. The main purpose of this note, is to move from one 4-cycle system to another one, using trades of volume 2 (which are called “double diamonds”). To do this, we use some linear algebraic techniques. We show that every trade is an element of the kernel of a special pair inclusion matrix M, and there exits a basis for the kernel of M consisting of just double diamonds