Device-to-Device communication allows a cellular user (relay node) to relay data between the base station (BS) and another cellular user (destination node). We address the problem of designing reverse auctions to assign a relay node to each destination node, when there are multiple potential relay nodes and multiple destination nodes, in the scenarios where the transmission powers of the relay nodes are: 1) fixed, 2) selected to achieve the data rates desired by destination nodes, and 3) selected so as to approximately maximize the BS's utility. We show that auctions based on the widely used Vickrey-Clarke-Groves (VCG) mechanism have several limitations in scenarios 1) and 2); also, in scenario 3), the VCG mechanism is not applicable. Hence, we propose novel reverse auctions for relay selection in each of the above three scenarios. We prove that all the proposed reverse auctions can be truthfully implemented as well as satisfy the individual rationality property. Using numerical computations, we show that in scenarios 1) and 2), our proposed auctions significantly outperform the auctions based on the VCG mechanism in terms of the data rates achieved by destination nodes, utility of the BS and/ or the interference cost incurred to the BS.