In this paper, the timeslot based traffic scheduling problem is considered with the objectives to fairly and energy efficiently serve the vehicles. The contact durations between the vehicles and the RSU are slotted into timeslots, during each of which one unit of the traffic can be transmitted from the RSU to one vehicle, and then one unit of vehicle's requirement can be satisfied. The downlink traffic scheduling cost is defined as the transmission energy cost to deliver the traffic. The fairness of the traffic scheduling strategy is evaluated by the Jain's fairness index of the unsatisfied vehicle requirements. The proposed problem is formulated as a three-step optimization problem. To address the difficulty of the three-step optimization problem, a flow network based algorithm is proposed. Both the offline and the online traffic schedule strategies are derived. Simulation results show that the derived offline traffic scheduling strategy outperforms the original energy optimal traffic scheduling strategy in terms of fairness, while the transmitted traffic and the transmission energy cost remains almost the same. The derived online traffic scheduling strategy outperforms the existing GMCF algorithms in terms of fairness, at the expense of larger transmission energy, and with the same total transmission traffic.