flapper
Try the flapper first......
Could it be that you have high water water pressure?
Check to see that you don't have so much water coming through the refill tube into the stand pipe that it forces water back towards the opening in the flush valve, keeping the flapper from closing.
If there is no flow adjustment on the refill tube and you have a high volume flow into the ballcock, cut a short piece of 1/8" tubing and insert it into the outlet of the refill tube, which will restrict the flow.
Then again this may not apply to you, but it's worth consideration.