Mobile Edge Computing (MEC) is a promising solution to improve vehicular services through offloading computation to cloud servers in close proximity to mobile vehicles. However, the self-interested nature together with the high mobility characteristic of the vehicles make the design of the computation offloading scheme a significant challenge. In this paper, we propose a new Vehicular Edge Computing (VEC) framework to model the computation offloading process of the mobile vehicles running on a bidirectional road. Based on this framework, we adopt a contract theoretic approach to design optimal offloading strategies for the VEC service provider, which maximize the revenue of the provider while enhancing the utilities of the vehicles. To further improve the utilization of the computing resources of the VEC servers, we incorporate task priority distinction as well as additional resource providing into the design of the offloading scheme, and propose an efficient VEC server selection and computing resource allocation algorithm. Numerical results indicate that our proposed schemes greatly enhance the revenue of the VEC provider, and concurrently improve the utilization of cloud computing resources. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.