This paper presents achievable regions in slotted ALOHA throughput both without and with network coding for one-relay twohop wireless networks between two end node groups. In this paper, there are no restrictions on the total traffic and the number of end nodes per group. It follows that the relay node will be generally involved with asymmetric bidirectional traffic. This paper derives closed-form expressions of the throughput and packet delay per group both without and with network coding from a theoretical perspective regardless of whether the buffer on the relay node is saturated or not. Furthermore, we show that the maximum throughput per group with network coding can be achieved at the boundary of the relay buffer saturation and unsaturation which is expressed as the solution of a polynomial equation in two group node traffics. As a result, we clarify the enhancement of the achievable region in slotted ALOHA throughput by applying network coding.