TITLE: $n$ people at a party and friendship
QUESTION [5 upvotes]: There are $n\geq 2$ people at a party. Each person has at least one friend inside the party. Show that it is possible to choose a group of no more than $\frac{n}{2}$ people at the party, such that any other person outside the group has a friend inside it.
This is just the definition of Dominating sets! I think I could use the  marriage theorem or Hall's theorem... the solution I know (without using it) involves dividing into spanning trees and coloring with one of two colors.

REPLY [4 votes]: You can partition the graph as $V = V_1 \cup V_2$, where each vertex in $V_1$ has at least as many neighbours in $V_2$ as in $V_1$ and vice versa. To prove this, take the partition with the maximum amount of crossing edges. If a vertex has more neighbours in its own subgraph, when you move it to the other side, the number of crosses will increase, a contradiction.
Now, we're done, since such a partition $V_1 \cup V_2$ clearly satisfies the conditions of the problem.