Configurations in 4-cycle systems (Q1889824)

A \(4\)-cycle system of order \(n\) is a collection of \(4\)-cycles which form a decomposition of the edge set of the complete graph on \(n\) vertices. There are four possible configurations obtained as union of two \(4\)-cycles. The paper deals with counting frequencies of the four configurations in \(4\)-cycle systems. It is shown that the configuration isomorphic to the complete bipartite graph \(K_{2,4}\) plays a special role. For each admissible value of \(n,\) a \(4\)-cycle system of order \(n\) avoiding this configuration is constructed.