Decompositions into 2-regular subgraphs and equitable partial cycle decompositions

AbstractTwo 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 m1,m2,…,mt if and only if n is odd, 3⩽mi⩽n for i=1,2,…,t, and m1+m2+…+mt=n2. Secondly, it is proved that if there exists partial decomposition of the complete graph Kn of order n into t cycles of lengths m1,m2,…,mt, then there exists an equitable partial decomposition of Kn into t cycles of lengths m1,m2,…,mt. 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