Let G be a bipartite graph, and prove that the complement of G is a perfect graph. the complement of bipartite graph is a graph where the 2 groups of vertices of the original graph turns into a clique, where there may be some edges between those 2 cliques.
Obviously, the clique number of $G^c$ is the maximum between the sizes of the 2 groups of vertices. unfortunately, I have not managed to prove that the chromatic number of $G^c$ is the maximum between those 2 groups as well. would like to get some help here. Thank you!
Note: I know that a graph G is perfect $\iff$ its complement is perfect, and because a bipartite graph is perfect obviously their complement are perfect as well, but I need to prove that the complement of bipartite graph is perfect without using this theorem.
There is a counter-example to your statement about clique number. Consider a tree with two vertices of degree $3$ and four leaves. It is bipartite as any other tree is. Both parts of this graph have cardinality $3$. So you suppose that the complement graph has clique number $3$, while the correct value is $4$, because all leaves become adjacent to each other. (Here we could also go further and consider disconnected bipartite graphs, where part cardinalities may even change up to ``rotation'' of some component.)
The independence number of a graph equals to the clique number of its complement graph. Therefore we need to make a conclusion about independence number of bipartite graph. It follows from Kőnig's theorem that $\alpha_0(G) + \alpha_1(G) = |G|$, where $\alpha_0(G)$ is the independence number of $G$ and $\alpha_1(G)$ is cardinality of the maximum matching in $G$. So clique number of the complement of the bipartite graph $G$ is $\omega(\overline{G}) = \alpha_0(G) = |G| - \alpha_1(G)$.
At the same time maximum matching gives us a way to color vertices of $G$ into $|G| - \alpha_1(G)$ colors such that any pair of non-adjacent vertices have different colors. Just color two ends of an edge of some certain maximum matching into the same color. Use different colors for different edges and give to each non-matched vertex its own color. This corresponds to a proper coloring of $\overline{G}$. So the chromatic number of complement of the bipartite graph $G$ is $\chi(\overline G) \le |G| - \alpha_1(G)$. One the other hand $\chi(\overline G) \ge \omega(\overline G) = |G| - \alpha_1(G)$. Therefore $\chi(\overline G) = \omega(\overline G) = |G| - \alpha_1(G)$.