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Intelligent vehicular systems and smart city applications are the fastest growing Internet of things (IoT) implementations at a compound annual growth rate of 30%. In view of the recent advances in IoT devices and the emerging new breed of IoT applications driven by artificial intelligence (AI), fog radio access network (F-RAN) has been recently introduced for the fifth generation (5G) wireless communications to overcome the latency limitations of cloud-RAN (C-RAN). We consider the network slicing problem of allocating the limited resources at the network edge (fog nodes) to vehicular and smart city users with heterogeneous latency and computing demands in dynamic environments. We develop a network slicing model based on a cluster of fog nodes (FNs) coordinated with an edge controller (EC) to efficiently utilize the limited resources at the network edge. For each service request in a cluster, the EC decides which FN to execute the task, i.e., locally serve the request at the edge, or to reject the task and refer it to the cloud. We formulate the problem as infinite-horizon Markov decision process (MDP) and propose a deep reinforcement learning (DRL) solution to adaptively learn the optimal slicing policy. The performance of the proposed DRL-based slicing method is evaluated by comparing it with other slicing approaches in dynamic environments and for different scenarios of design objectives. Comprehensive simulation results corroborate that the proposed DRL-based EC quickly learns the optimal policy through interaction with the environment, which enables adaptive and automated network slicing for efficient resource allocation in dynamic vehicular and smart city environments.