Graph theory proof

1. Jan 14, 2016

TheMathNoob

1. The problem statement, all variables and given/known data
Exercise 0.1. Suppose that G is a finite graph all of whose vertices has degree two or greater. Prove that a cycle passes through each vertex. Conclude that G cannot be a tree.

2. Relevant equations

3. The attempt at a solution
If every vertex in a graph G has degree two or greater, then there is always a way to get in and out without using the same edge. If we start at any vertex and try to traverse the graph by using the rule of getting in and out and not using the same edge, eventually we will realize that we can't stop, so this implies that there is a cycle. Therefore G can't be a tree.

I am new to this types of proofs, so I need feedback.

Last edited: Jan 14, 2016
2. Jan 15, 2016

Samy_A

Consider the following graph:

Apply your proof to vertex 1. Does it show that there is a cycle through vertex 1?

3. Jan 15, 2016

haruspex

That's not just a counterexample to the proof, it's a counterexample to the given claim.

4. Jan 15, 2016

Samy_A

I know, it was a not too subtle hint.
The "each vertex" in the statement of the exercise must be an error, and the exercise was probably something like: "prove that the graph contains a cycle".

5. Jan 15, 2016

haruspex

Yes, that makes sense for the last part.