Device-to-Device (D2D) communications and cloud radio access network (C-RAN) can improve Spectrum Efficiency (SE). However, extremely severe intra-cell interference and inter-cell interference hinder the improvement of energy efficiency (EE). In the existing typical work, a centralized interference mitigation algorithm is deployed in cellular infrastructure to try to eliminate these types of interferences. Although this method improves the quality of service, it hardly reduces the energy consumption of cellular user equipments (UEs). Moreover, it increases the energy consumption of cellular infrastructure. In this paper, we first introduce out-band D2D relays to assist the cellular communications, which can shorten the average transmission distance of cellular UEs and thus make them improve their EE and reduce the inter-cell interference for each other. Then, we propose an energy-efficient resource sharing scheme to determine channel selection and power allocation. Next, we formulate the resource sharing problem as the noncooperative game model, where each UE optimizes its EE respectively with the aid of the remote radio heads in C-RAN. Finally, in order to obtain the optimal EE of each UE, we let the non-concave optimization problem be transformed into the concave form by using constraint relaxation and nonlinear fractional programming and solve the transformed problem by Dinkelbach's method and the Lagrangian duality theory. The simulation results show that the SE and EE of cellular UEs can be improved by 64.63% and 24.97%, respectively when adopting the parameters with the best values in our relay selection strategies. INDEX TERMS Cloud radio access network, out-band D2D relay-aided communication, noncooperative game, resource sharing, energy efficiency.