Decomposition of hypercubes into sunlet graphs

For an integer \(k \ge 3,\) $k\ge 3,$ the sunlet graph of order 2k, denoted by \(L_{2k},\) $L_{2k},$ is a graph obtained from a cycle of length k by attaching one pendant vertex to each vertex of the cycle. The cycle decompositions of the hypercubes is well studied in the literature. In this paper, we obtain sunlet decompositions of hypercubes using cycle decompositions. We prove that if the hypercube \(Q_{2n}\) $Q_{2n}$ has a decomposition into cycles of length k,  then \(Q_{mn}\) $Q_{\mathit{mn}}$ has a decomposition into copies of \(L_{2k}\) $L_{2k}$ for \(m=4\) $m=4$ or \(m \ge 6.\) $m\ge 6.$ Further, we obtain a necessary and sufficient condition for the existence of a decomposition of \(Q_n\) $Q_n$ into spanning sunlet graphs. As a consequence, we obtain the decomposition of \(Q_{mn}\) $Q_{\mathit{mn}}$ into sunlet graphs of various orders using cycle decomposition. In particular, we show that \(Q_n\) $Q_n$ has an \(L_{q}\) $L_q$ -decomposition for \(q = 8\) $q=8$ and \(q=16\) $q=16$ whenever \( n = 4\) $n=4$ or \( n \ge 6.\) $n\ge 6.$