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A twofold 8-cycle system is an edge-disjoint decomposition of a twofold complete graph (which has two edges between every pair of vertices) into 8-cycles. The order of the complete graph is also called the order of the 8-cycle system. A twofold 2-perfect 8-cycle system is a twofold 8-cycle system such that the collection of distance 2 edges in each 8-cycle also cover the complete graph, forming a (twofold) 4-cycle system. Existence of 2-perfect 8-cycle systems for all admissible orders was shown in [1], although \(\lambda\)-fold existence for \(\lambda > 1\) has not been done.
In this paper, we impose an extra condition on the twofold \(2\)-perfect \(8\)-cycle system. We require that the two paths of length two between each pair of vertices, say \(x, a_{xy}, y\) and \(x, b_{xy}, y\), should be distinct, that is, with \(a_{xy} \neq b_{xy}\); thus they form a \(4\)-cycle \((x, a, y, b)\).
We completely solve the existence of such twofold 2-perfect 8-cycle systems with this “extra” property. All admissible orders congruent to $0$ or 1 modulo 8 can be achieved, apart from order 8.