Sandbox:DavidKohler/Alon try 1/Theorem 3.9

Statement

Theorem 3.9
Let G be a connected graph, then the following are equivalent: (1) The graph G is 1-loopy. (2) The corresponding graph of irreducible walks GIrred is strongly connected. (3) The graph G is not a cycle and all its vertices have degree at least 2. Furthermore, if G is also d-regular (for d ≥ 3) then the graph GIrred is strongly connected.

Proof

We'll do a cyclic proof.

(1)=>(2)

Let

(2)=>(3)

If G is a cycle, then.

(3)=>(1)

Let e be any edge.