The aim of this paper is to present a groundwork on the delay-minimized routing problem in a vehicular ad-hoc network (VANET) where some of the vehicles are equipped with full-duplex (FD) radios. We first give the generalized delay calculation model for a multi-hop path, and prove that the Dijkstra algorithm is unable to get the delay-minimized routing path from source to destination. Then we propose two routing methods: graph-based method and deep reinforcement learning (DRL)-based method. In the graph-based method, the network topology is reformulated as an equivalent graph and then an evolved-Dijkstra algorithm is proposed. In the DRL-based method, the deep Q network (DQN) is employed to learn the shortest end-to-end path, wherein the delay is modeled as the rewards for routing actions. The graph-based method can achieve the exact minimum end-to-end delay, while the DRL-based method is more feasible due to its acceptable complexity. Finally, extensive simulations demonstrate that the DRL-based approach with proper hyper-parameters can achieve near minimum end-to-end delay, and the achieved delay has a notably decline as the number of FD nodes increases.