Definition:A connected graph that contains no cycles is called a TREE.

Theorem:(This is a theorem that I assume you know). Given a graph that is a tree the # of edges is 1 less the # of vertices.

Now given any (finite) graph we can break it up into connected peices. Those peices are called CONNECTED COMPONENTS. So for example an ordinary tree has 1 connected components.

Definition:A graph where each connected component is a tree is called a FOREST.

Using the theorem that was mentioned try to prove the following stronger result. Given a graph with connected components with edges and vertices then .

Now it is pretty obvious that if then there must exist a cycle.