$Q_4$-Factorization of $\lambda K_n$ and $\lambda K_x(m)$

Abstract

In this study, we show that necessary conditions for $Q_4$-factorization of $\lambda{K_n}$ and $\lambda{K_{x(m)}}$ (complete $x$ partite graph with parts of size $m$) are sufficient. We proved that there exists a $Q_4$-factorization of $\lambda{K_{x(m)}}$ if and only if $mx\equiv{0} \pmod{16}$ and $\lambda{m(x-1)}\equiv{0}\pmod{4}$. This result immediately gives that $\lambda K_n$ has a $Q_4$-factorization if and only if $n\equiv 0 \pmod{16}$ and $\lambda \equiv 0 \pmod{4}$.