Q4-factorization of lambda λ Kn and lambda λ Kx(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 equivalent to 0 (mod 16) and lambda m(x - 1) equivalent to 0 (mod 4). This result immediately gives that lambda Kn has a Q(4)-factorization if and only if n equivalent to 0 (mod 16) and lambda equivalent to 0 (mod 4).