Decomposing certain equipartite graphs into sunlet graphs of length 2p

For any integer r≥3, we define the sunlet graph of order 2r, denoted L2r, as the graph consisting of a cycle of length r together with r pendant vertices, each adjacent to exactly one vertex of the cycle. In this paper, we give necessary and sufficient conditions for decomposing the lexicographic product of the complete graph and the complete graph minus a 1-factor, with complement of the complete graph Km, (that is Kn⊗K̄m and Kn−I⊗K̄m, respectively) into sunlet graphs of order twice a prime.