In a motorcycle, a head pipe is provided with one main frame part that extends rearwardly from the head pipe and a rear frame that extends substantially downwardly from the rear of the main frame part. A pivot is pierced in a pivot support part provided to the rear frame with a swing arm being attached to the pivot so that the swing arm can be vertically swung and a rear shock absorber unit is positioned between the swing arm and the side of the body frame. The main frame is cast in an aluminum alloy and the rear frame on the upside and the downside of the pivot supporting part projects on both sides of a vehicle from the pivot supporting part to attach right and left brackets for supporting both ends of the pivot.