I received the following statement:

"If a graphic $ G $ contains no odd circuits $ C_ {2k + 1} $ to the $ k geq $ 2or his supplement, then we have $ omega (G & # 39;) alpha (G & # 39;) geq | V (G & # 39;) | $ for every induced subgraph $ G & # 39; $ from $ G $ "

from where $ omega $ is the clique number, $ alpha $ is the stability number, $ | V | $ is the number of vertices in the graph. This actually shows that if we prove the strong perfect graph set, we already prove the perfect graph set.

I'm trying to prove the claim, and I can show that if there's a cycle $ C_ {2k + 1} $ With $ k geq $ 2, we have $ alpha (C_ {2k + 1}) = k, omega (C_ {2k} +1) = 2 $, The multiplication becomes $ 2k <2k + 1 $ (similar to the supplement). My questions are however:

- How can I show this if a subgraph is not $ C_ {2k + 1} $ then we have the condition?
- Is there a difference if I use the circuit instead of the cycle?