If you pair everyone off, either all three groups are disjoint, or some of the groups are connected. If you don't pair people, you end up with 2 completely dis

If you pair everyone off, either all three groups are disjoint, or some of the groups are connected. If you don't pair people, you end up with 2 completely disconnected people and also at least one more person that's not connected to either Obviously if you make a group bigger than a pair, you also end up with 3 connected people