Decompositions of complete graphs into blown-up cycles [formula omitted

Abstract

Let C m [ K ¯ 2 ] stand for a cycle C m in which every vertex is replaced by two isolated vertices and every edge by K 2 , 2 . We prove that the complete graph K 8 m k + 1 can be decomposed into graphs isomorphic to C m [ K ¯ 2 ] for any m ≥ 3 , k > 0 . Decompositions of complete graphs into certain collections of even cycles are obtained as a corollary. Also some special cases of Alspach Conjecture are solved in this article. All proofs are constructive and use both graph theory and design theory techniques.