Decompositions into 2-regular subgraphs and equitable partial cycle decompositions

Two theorems are proved in this paper. Firstly, it is proved that there exists a decomposition of the complete graph of order n into t edge-disjoint 2-regular subgraphs of orders m 1 , m 2 , … , m t if and only if n is odd, 3 ⩽ m i ⩽ n for i = 1 , 2 , … , t , and m 1 + m 2 + … + m t = n 2 . Secondly, it is proved that if there exists partial decomposition of the complete graph K n of order n into t cycles of lengths m 1 , m 2 , … , m t , then there exists an equitable partial decomposition of K n into t cycles of lengths m 1 , m 2 , … , m t . A decomposition into cycles is equitable if for any two vertices u and v, the number of cycles containing u and the number of cycles containing v differ by at most 1.