In an ad hoc network, each node can be in one of three states: asleep (powered down), listening, or transmitting. Communication is effective only when the sender is transmitting, the destination is receiving, and no other nodes in proximity to the receiver are also transmitting. Our strategy makes no assumptions about knowledge of neighbours or of geographical position; it is topology-transparent. A general combinatorial model for topology-transparent scheduling that treats energy conservation is described. As in the two state (transmit and receive) case, the combinatorial requirements are met by a D cover-free family. Graph designs, where an arc from vertex x to y indicates an opportunity for x to transmit and y to receive, are proposed as a model for schedule construction. In order to achieve reasonable throughput while obtaining a dramatic reduction in energy consumption, we focus on Ka,a → designs, where the number of nodes transmitting and receiving per slot is equal to a. Patterned on constructions for resolvable designs, we examine a computational search method to meet the required combinatorial conditions.