Decomposition of Regular Bipartite Graphs Into Hamiltonian Cycles (Paths) and S3

Let G be either a complete bipartite graph with n (even) vertices in each partite or a complete bipartite graph with n (odd) vertices in each partite plus a 1†factor. A Hamiltonian cycle (respectively, path) of G is a cycle (respectively, path) that visits each vertex exactly once. In this paper, we determine the necessary and sufficient conditions for decomposing the graph G into λ copies of Hamiltonian cycles (or paths) and μ copies of the S3, a star with three edges if and only if n2+εn=2nλ+3μor n2+εn=2n−1λ+3μ, where ε=1 if n is odd and ε=0 if n is even.