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This paper studies the subchannel and power allocation problem in the relay-enhanced downlink orthogonal frequency-division multiple access (OFDMA) systems. The challenges of this resource allocation problem arise from the complication of two-phase assignments within a subchannel since the relay station (RS) can provide an additional two-hop signal path from the base station to the user equipments (UEs). Existing research work does not fully consider all the influential factors to achieve feasible resource allocation for the relay-based networks. In this paper, the QoS-based resource allocation (QRA) schemes are proposed to design the subchannel and power allocations for the UEs with the consideration of direct and two-hop communications. Both the selective and fixed phase assignments for the UEs are addressed to obtain the suboptimal solution for the proposed QRA schemes. Different weights are designed for the UEs to exploit the fairness and maximization of system throughput for the relay-enhanced networks. Simulation results show that the proposed QRA schemes with selective phase assignment can provide comparably higher network throughput with QoS consideration.