Graph Theory: Prove Tree Has No Cycles & Adding Edge Creates Cycle

[pupeye11]

Homework Statement

Prove that a graph is a tree if and only if it has no cycles and the insertion of any new edge always creates exactly one cycle.

The Attempt at a Solution

Assume that a graph G is connected and contains no vertices with a degree of zero.

So would I get my proof by proving that the graph is a tree and it is connected and has n-1 edges which proves it has no cycles and then prove that adding a edge would create a cycle making the first statement false. Does this work?

 

[Tedjn]

Two directions. In the first, you have a tree. Facts about trees: they are connected and acyclic. It is a fact that a tree has exactly n-1 edges, but depending on what you are allowed to assume, you may need to prove this. Starting from these facts, you must prove that adding any additional edge will create a cycle. In the second, you have a graph such that the insertion of any edge creates a cycle. You must prove that this graph is a tree, i.e. connected and acyclic.