On Decompositions of Complete Multipartite Graphs into the Union of two Even Cycles

For positive integers \(c\) and \(d\), let \(K_{c\times d}\) denote the complete multipartite graph with \(c\) parts, each containing \(d\) vertices. Let \(G\) with \(n\) edges be the union of two vertex-disjoint even cycles. We use graph labelings to show that there exists a cyclic \(G\)-decomposition of \(K_{(2n+1)\times t}\), \(K_{(n/2+1)\times 4t}\), \(K_{5\times (n/2)t}\), and of \(K_{2\times 2nt}\) for every positive integer \(t\). If \(n \equiv 0 \pmod{4}\), then there also exists a cyclic \(G\)-decomposition of \(K_{(n+1)\times 2t}\), \(K_{(n/4+1)\times 8t}\), \(K_{9\times (n/4)t}\), and of \(K_{3\times nt}\) for every positive integer \(t\).