Maxiumum packing of inside perfect 8-cycle systems

Abstract

For an m-cycle C, an inside m-cycle of C is a cycle that is on the same vertex set and edge-disjoint from C. In an m-cycle system, (X, C), if inside m-cycles can be chosen-one for each cycle-to form another m-cycle system, then (chi, C) is called an inside perfect m-cycle system. Inside perfect cycle systems can be considered as generalisations of i-perfect cycle systems. Cycle packings are generalisations of cycle systems that may have leaves after decomposition. In this paper, we prove that an inside perfect maximum packing of K-n with 8-cycles of order n exists for each n >= 8. We also construct a maximum 8-cycle packing of order n which is not inside perfect for each n >= 10.