Hey I'd love a bit of help with this question....

Prove that if G is a triangle-free graph or order n and δ(G) > 2n/5 then G must be bipartite.

I've started thus far...

Suppose not, then G contains an odd cycle C of length 2k+1 with k >= 2. I want to show there is a vertex v that is adjacent to at least three vertices of C, but I can't. Any suggestions?