\((G_{m}, H_{m})\)-Multifactorization of \(\lambda K_{m}\)

A \((G,H)\)-multifactorization of \(\lambda K_m\) is a partition of the edge set of \(\lambda K_m\) into \(G\)-factors and \(H\)-factors with at least one \(G\)-factor and one \(H\)-factor. Atif Abueida and Theresa O’Neil have conjectured that for any integer \(n \geq 3\) and \(m \geq n\), there is a \((G_n, H_n)\)-multidecomposition of \(\lambda K_m\) where \(G_n = K_{1,n-1}\) and \(H_n = C_n\). In this paper, it is shown that the above conjecture is true for \(m=n\) when

1. \(G_m = K_{1,m-1}; H_m = C_m\),
2. \(G_m = H_{1,m-1}; H_m = P_m\), and
3. \(G_m = P_m; H_m = C_m\).