I want to show for a Combinatorial Optimization lecture that bipartite graphs doesn't have cycles of odd length.
I think the idea is to prove that a bipartite graph doesn't have cycles by recurrence.
let's suppose that $G$ is a bipartite graph of even length for SOMETHING set. Let's show that the FOLLOWING SOMETHING is also a bipartite graph of even length.
Yet, what is the something to use as far as length can't be used... ? I think it is the number of vertex. A bipartite graph is bipartite if and only if it is 2-colorable (Wikipedia) but I'm not sure I'm going in the good way... Can you help me proving that bipartite graphs doesn't have cycles of odd length ?
I know a that a proof exists in Asratian, Armen S.; Denley, Tristan M.J.; H\"aggkvist, Roland, Bipartite graphs and their applications, Cambridge Tracts in Mathematics. 131. Cambridge: Cambridge University Press. xi, 259 p. (1998). ZBL0914.05049.