A Reduction Of The Spectrum Problem For Odd Sun Systems And The Prime Case

A k-cycle with a pendant edge attached to each vertex is called a k-sun. The existence problem for k-sun decompositions of K-v, with k odd, has been solved only when k = 3 or 5. By adapting a method used by Hoffmann, Lindner, and Rodger to reduce the spectrum problem for odd cycle systems of the complete graph, we show that if there is a k-sun system of Kv (k odd) whenever v lies in the range 2kv6k and satisfies the obvious necessary conditions, then such a system exists for every admissible v >= 6k. Furthermore, we give a complete solution whenever k is an odd prime.