First we will define some basic terminologies.

**Bipartite Graph:** A graph is bipartite if we can spit the vertices of the graph in two distinct sets V1 and V2 such that no edge connects vertices belonging to the same set.

**Hamiltonian Graph:** A graph is hamiltonian if it contains a hamiltonian circuit. Hamiltonian circuit is the circuit that visits each vertex of the graph exactly once without repetition.

**Proof that a bipartite graph with an odd number of vertices is not Hamiltonian**

We will prove the above statement by contradiction. Suppose a bipartite graph G with an odd number of vertices. Let G be hamiltonian. If G is hamiltonian then there exists a cycle in the graph that visits each vertex exactly once. The hamiltonian cycle will have an odd number of vertices. But this is not possible. A bipartite graph cannot contain an odd cycle. Therefore, our assumption that G is hamiltonian is wrong. Therefore, we can say that a bipartite graph with an odd number of vertices is not hamiltonian.

You can find proof that a bipartite graph doesn’t contain odd cycle here.