With the increasing demand for advanced autonomous driving, the available communication resources may become constrained over different geographic areas. In addition, due to dynamic channel variations and imperfect cell deployments, guaranteeing the required communication resources for data hungry and delay-sensitive applications in autonomous vehicles (AVs), along their entire trips, becomes challenging. To address these issues, the paper investigates the feasibility of a hybrid system-optimum and user-equilibrium AV traffic framework subject to communication constraints, as well as its performance gain. Within such a framework, the paper introduces the problems of communication-constrained routing (CCR) and traffic control (CCTC) in the context of infrastructure-assisted autonomous driving and presents respective solutions. For CCR, an efficient two-layered routing scheme is proposed which can provide optimal trip duration. Simulation results show that the routing scheme achieves a good balance between longer duration of communication coverage and acceptable source-to-destination travel time. For CCTC, it is shown that there exists an optimal AV speed on each road segment, as well as an optimal inter-AV distance and an optimal number of AVs in each cell, to maximize the road-network AV throughput within a single cell. Moreover, spectrum allocation is used to achieve Pareto-optimal road-network throughput across cells, and a new key performance index (KPI) is defined to evaluate the traffic control capability of cellular systems. Simulation results validate the improvement of AV throughput via the proposed CCTC solution.