The extended metamorphosis of a complete bipartite design into a cycle system

Abstract

A K t, t -design of order n is an edge-disjoint decomposition of K n into copies of K t, t . When t is odd, an extended metamorphosis of a K t, t -design of order n into a 2 t-cycle system of order n is obtained by taking ( t−1)/2 edge-disjoint cycles of length 2 t from each K t, t block, and rearranging all the remaining 1-factors in each K t, t block into further 2 t-cycles. The ‘extended’ refers to the fact that as many subgraphs isomorphic to a 2 t-cycle as possible are removed from each K t, t block, rather than merely one subgraph. In this paper an extended metamorphosis of a K t, t -design of order congruent to 1 ( mod 4t 2) into a 2 t-cycle system of the same order is given for all odd t>3. A metamorphosis of a 2-fold K t, t -design of any order congruent to 1 ( mod t 2) into a 2 t-cycle system of the same order is also given, for all odd t>3. (The case t=3 appeared in Ars Combin. 64 (2002) 65–80.) When t is even, the graph K t, t is easily seen to contain t/2 edge-disjoint cycles of length 2 t, and so the metamorphosis in that case is straightforward.