The Metamorphosis of \(K_4 \backslash e\) Designs Into Maximum Packings of \(K_n\) With \(4\)-Cycles

Abstract

Let \(K_4\backslash e=…\). If we remove the “diagonal” edge, the result is a \(4\)-cycle. Let \((X,B)\) be a \(K_4\backslash e\) design of order \(n\); i.e., an edge-disjoint decomposition of \(K_n\) into copies of \(K_4\backslash e\). Let \(D(B)\) be the collection of “diagonals” removed from the graphs in \(B\) and \(C(B)\) the resulting collection of \(4\)-cycles. If \(C_2(B)\) is a reassembly of these edges into \(4\)-cycles and \(L\) is the collection of edges in \(D(B)\) not used in a \(4\)-cycle of \(C_2(B)\), then \((X, (C_1(B) \cup C_2(B)), L)\) is a packing of \(K_n\) with \(4\)-cycles and is called a metamorphosis of \((X,B)\). We construct, for every \(n = 0\) or \(1\) (mod \(5\)) \(> 6\), \(n \neq 11\), a \(K_4\backslash e\) design of order \(n\) having a metamorphosis into a maximum packing of \(K_n\) with \(4\)-cycles. There exists a maximum packing of \(K_n\) with \(4\)-cycles, but it cannot be obtained from a \(K_4\backslash e\) design.