I am trying to teach myself graph theory. I am using Diestel's book found at http://diestel-graph-theory.com/ . Anyway, Diestel states in proposition 1.2.2:

Every graph G with at least one edge has a subgraph H with delta(H) > epsilon(H) >= epsilon(G) .

I am attempting to make an example but I cannot make it work. Here are my steps, can you tell me where I am wrong:

1. Graph G has 2 vertices and 1 edge: *------*

2. Graph H has 1 vertex: *

3. The delta(H) = the minimum degree of G = min{d(v), where v is an element of V}

4. I calculate the delta(H) = 0

5. epsilon(H) = The number of edges of a H per vertex expressed as |E| / |V|.

6. I calculate epsilon (H) = 0/1 = 0

7. epsilon (G) = 1/2

8. but, 0 > 0 >= 1/2 is not true.

Where am I going wrong?

Thanks for any help,

Carlos