On Cyclic \(G\)-Designs where \(G\) is the One-Point Union of Two Cycles

Let \( G \) be the one-point union of two cycles and suppose \( G \) has \( n \) edges. We show via various graph labelings that there exists a cyclic \( G \)-decomposition of \( K_{2nt+1} \) for every positive integer \( t \).