The extended metamorphosis of a complete bipartite design into a cycle system Original Research Article

A Kt,t-design of order n is an edge-disjoint decomposition of Kn into copies of Kt,t. When t is odd, an extended metamorphosis of a Kt,t-design of order n into a 2t-cycle system of order n is obtained by taking (t−1)/2 edge-disjoint cycles of length 2t from each Kt,t block, and rearranging all the remaining 1-factors in each Kt,t block into further 2t-cycles. The ‘extended’ refers to the fact that as many subgraphs isomorphic to a 2t-cycle as possible are removed from each Kt,t block, rather than merely one subgraph. In this paper an extended metamorphosis of a Kt,t-design of order congruent to 1 (mod 4t2) into a 2t-cycle system of the same order is given for all odd t>3. A metamorphosis of a 2-fold Kt,t-design of any order congruent to 1 (mod t2) into a 2t-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 Kt,t is easily seen to contain t/2 edge-disjoint cycles of length 2t, and so the metamorphosis in that case is straightforward.