\(P_3\)-factorization of complete multipartite graphs (Q1288299)

The author provides necessary and sufficient conditions for the existence of a \(P_3\)-factorization of the complete multipartite graph \(\lambda K^n_m\). Namely he proves such conditions are (i) \(m\geq 3\), (ii) \(mn\equiv 0\pmod 3\), and (iii) \(\lambda(m-1)n\equiv 0\pmod 4\). The proof is by construction and uses suitable resolvable group divisible designs with an even number of resolution classes.