Minimum? You can put one frame of brood and one frame of honey in a two frame nuc box and they will raise a queen. The quality of the queen depends on several things, though, and the density of the bees and availability of resources (pollen and nectar) have a marked effect. Also, a nuc that small will not build up quickly.
Three frames of brood and two frames of honey/pollen with about three more frames of open brood shaken into the box (the box should be FULL of bees) will make a nice split that will, after they get a new queen laying, build up fairly quickly.
http://www.bushfarms.com/beessplits.htm