Welcome to the forums!
You have made some graphic designs and someone else want to purchase them, put them on shirts, and sell them?
I will take it that your answer is yes and go from there. Correct me if I am wrong.
This will depend on what you think your designs are worth, first of all and what they are willing to pay for them.
Just starting out, royalties are pretty rare and also hard to keep track of. So you must consider that they are going to want to come out of this the cheapest and least complicated way they can, which is probably a one time fee to purchase your designs outright.
Have you thought about having your designs put on shirts yourself. There's more grunt work, but there's quite a few of us here that find it rewarding. Also, you get to keep the money and don't have to worry about royalties and everything that goes with it.
Depending on how hands on you want to be there are several options available to you if you wanted to sell shirts with your designs yourself.
Back to your question. If you still decided to go the route of selling your designs, I would strongly suggest selling them outright. How would you know how many shirts were sold in order to get your cut, if you did it on a royalty basis? There's just too many problems and too much hassle doing it that way IMO.
