Need help on a combinatorics problem!

we call somebody "aloof" that his friends are less than 10...and we call somebody "strange" that all his friends are "aloof" ... if the number of "aloof" ones are m and the number of "strange" ones are n ... prove that m is bigger/equal than n.

Question

Need help on a combinatorics problem!

 we call somebody "aloof" that his friends are less than 10...and we call somebody "strange" that all his friends are "aloof" ... if the number of "aloof" ones are m and the number of "strange" ones are n ... prove that m is bigger/equal than n.

anonymous · Answer

@SithsAndGiggles please write your answer.

anonymous · Answer

A person is "aloof" is they have less than 10 friends. Suppose they have the maximum number of friends, 9. Then you can model an aloof person with the graph
|dw:1405993836397:dw|
If you're not familiar with graphs, here's a basic definition. A graph is a set $G(V,E)$ consisting of two sets, $V$ and $E$, which denote vertices and edges, respectively. Vertices (dots in the picture) are connected by edges (line segments in the picture).

So an "aloof" person is one such vertex that is incident to at most 9 other vertices. (Vertices A and B are incident to each other if there is an edge connecting A and B.)