In this paper the authors give necessary and sufficient conditions for the oscillation of solutions of nonlinear delay difference equations of Emden-Fowler type in the form $\Delta^2y_{n-1}+q_ny_{\sigma(n)}^\gamma=g_n$, where $\gamma$ is a quotient of odd positive integers, in the superlinear case $(\gamma>1)$ and in the sublinear case $(\gamma<1)$.