TITLE: Graph theory , a problem
QUESTION [6 upvotes]: I am trying to solve the following problem:

In a non-empty graph, every two vertices with the same degree do not
  have a common neighbor. Prove that there exists a vertex with degree
  one in this graph.

REPLY [3 votes]: Let $v$ be a vertex of maximal degree $k$.
The $k$ neighbours of $v$ must all have a different degree, otherwise $v$
itself is common neighbour of two vertices with the same degree.
So the neighbours of $v$ must have degrees $1,2,\ldots,k$.
This exhibits a vertex of degree 1.