Mobile-edge computation offloading (MECO) is an emerging technology for enhancing mobiles' computation capabilities and prolonging their battery lives, by offloading intensive computation from mobiles to nearby servers such as base stations. In this paper, we study the energy-efficient resource-management policy for the asynchronous MECO system, where the mobiles have heterogeneous input-data arrival time instants and computation deadlines. First, we consider the general case with arbitrary arrival-deadline orders. Based on the monomial energy-consumption model for data transmission, an optimization problem is formulated to minimize the total mobile-energy consumption under the time-sharing and computation-deadline constraints. The optimal resource-management policy for data partitioning (for offloading and local computing) and time division (for transmissions) is shown to be computed by using the block coordinate decent method. To gain further insights, we study the optimal resource-management design for two special cases. First, consider the case of identical arrival-deadline orders, i.e., a mobile with input data arriving earlier also needs to complete computation earlier. The optimization problem is reduced to two sequential problems corresponding to the optimal scheduling order and joint data-partitioning and time-division given the optimal order. It is found that the optimal time-division policy tends to balance the defined effective computing power among offloading mobiles via time sharing. Furthermore, this solution approach is extended to the case of reverse arrival-deadline orders. The corresponding time-division policy is derived by a proposed transformation-and-scheduling approach, which first determines the total offloading duration and data size for each mobile in the transformation phase and then specifies the offloading intervals for each mobile in the scheduling phase.