A study of $4-$cycle systems

• DOI: 10.48550/arxiv.2308.10872
• Publication Date: 2023-01-01
• Authors: Gh., B. Bagheri; Khosravi, M.; Mahmoodian, E. S.; Rashidi, S.
• Keywords: 05B30, 05B05; FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO)

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.