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 + 3μ(or n2 + εn = (2n − 1)λ + 3μ), where ε = 1 if n is odd and ε = 0 if n is even.