Repressilators are biological regulatory networks in which components interact only in terms of negative influences. They are of interest in biology, since their oscillatory behavior can inform the design of gene therapies. Although sustained oscillations are ensured in 3-dimensional repressilators, that is, in systems made of 3 genes, they do not always appear in higher dimensions, as their occurrence depends on the topology of the network and on the chosen parameters. Here we focus on discrete models, where the presence of at least one cyclic attractor is required for sustained oscillations. Even in a discrete framework, enumerating and simulating all possible models can quickly become computationally infeasible. In this paper, we provide a sufficient condition for the presence of sustained oscillations for a class of repressilators, based on the structure of their influence graphs. The condition applies in any dimension and independently of the parameters, that is, threshold labeling of the edges. We also study the coexistence of cyclic attractors and fixed points in dimension 4.