Which graphs embed in the plane? We have proved that K₅ and K_3,3 donot. In fact, these are the crucial graphs and lead to a characterization  of planar graphs known as Kuratowski's Theorem. Kasimir Kuratowski once asked  Frank Harary about the origin of the notation for K5 and K3,3. Harary replied,  '''The K in K5 stands for Kasimir, and the K in K3,3 stands for Kuratowski!" Recall that a subdivision of a graph is a graph obtained from it by replacing  edges with pairwise internally-disjoint paths

Theorem 1.1.

 A graph is bipartite if and only if it contains no odd cycle.

Proof (outline). The proof of the necessary condition is easy when reasoning by the absurd and, in relation to the classes of the bipartition, following in order the vertices of an odd cycle. The proof of the sufficient condition isless simple but can be done in a constructive way, that is by producing the adequate bipartition. The principle is as follows: mark a first arbitrarily chosen vertex 0, then mark its neighbors 1, then take each of the newly marked vertices and mark their not-yet-marked neighbors 0, and so on until all vertices reached are marked 0 or 1. The crucial point is that if during this marking process two neighboring vertices happen to receive the same mark (twice 0 or twice 1) then there is an odd cycle in the graph. This can be seen by considering the paths defined by the succession of marked vertices which come to these two vertices and the edge joining them. With the hypothesis of the sufficient condition, this circumstance of two neighboring vertices bearing the same mark will not occur. The marks given to the vertices will define a bipartition in compliance with the definition of bipartite graphs. Any vertex will be marked as soon as the graph is connected, otherwise we should proceed independently with each connected component.