Suppose I have a number of airports which have a set number of routes. Large airports have 7, medium airports have 4, and small airports have 1.
First of all, what would I call a graph of this type where it is undirected, has no parallel edges, no loops, and every edge is connected to another (that is no route goes unused)?
Second of all, how can I determine if I get a number number of airports if it'll work / is connected? Example: 1 large, 3 medium, and 7 small will produce a connected graph but that same combination with 8 small will not.
I'm writing a computer program and looking for a simple way to determine (using some theorem or whatnot) if it works or not. I've attached an image below of what I'm searching for.