Diagonally Switchable \(\lambda\)-Fold \(4\)-Cycle Systems, \(\lambda > 1\)

Abstract

A diagonally switchable \(\lambda\)-fold \(4\)-cycle system of order \(n\), briefly DS4CS\((n, \lambda)\), is a \(\lambda\)-fold \(4\)-cycle system in which by replacing each \(4\)-cycle \((a,b,c,d)\) covering pairs \(ab, bc, cd, da\) by either of the \(4\)-cycles \((a,c,b,d)\) or \((a,d,c,b)\) another \(\lambda\)-fold \(4\)-cycle system is obtained. In \([3]\) Adams, Bryant, Grannell, and Griggs proved that a DS4CS\((n, 1)\) exists if and only if \(n \equiv 1 \pmod{8}\), \(n \geq 17\) with the possible exception of \(n = 17\). In this paper we prove that for \(\lambda \geq 2\) the necessary conditions for the existence of a \(A\)-fold \(4\)-cycle system of order \(7\) are also sufficient for the existence of a DS4CS\((n, \lambda)\) except for \((n, \lambda) = (5, 2)\).