$(G_m,H_m)$-Multifactorization of $\lambda K_m$

Abstract

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 $n\ge 3$ and $m\ge n$, 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. $G_m=P_m;H_m=C_m$.