There are so many great things you can do with a PVC pipe. But if the pipe isn't fitting and you need to cut it without tools, then nylon rope or twine will do the trick.
A Concord Carpenter recommends 20 inches of string, which you will need to pass through to the other side of the pipe. Grab both ends of the string — preferably use pieces of wood or a couple of pens to wrap both ends, so that your hands get a better grip. Make sure the twine is taught.
Pull one hand, then pull the other, then pull the first, and keep going with this repetitive movement as quick as you can. If the pipe is moving too much, hold it in place with your foot or some other object. In a short while, the twine will cut through the pipe clean.