Summary: The number of possibilities for random dating
and Rod Canfield
and Andrew Granville
Let G be a regular graph and H a subgraph on the same vertex set.
We give surprisingly compact formulas for the number of copies of H one
expects to find in a random subgraph of G.
There are n boys and n girls who apply to a computer dating service, which
randomly picks a boy and a girl and then introduces them. It does this again
and again until everyone of the boys and girls has been introduced to at least one
other person. The service then organizes a special evening at which everyone
dates someone to whom they have been previously introduced. In how many
different ways can all of the boys and girls be matched up?
Translating this question into the language of graph theory, we select edges
at random from the complete bipartite graph Kn,n until the subgraph created
by these edges, G, has minimum degree 1. We then ask how many perfect
matchings (subgraphs on the same vertex set where every vertex has degree 1)
are contained in G?