Bipartite Graphs

A bipartite graph is a graph whose vertex set can be split into two disjoint sets, V₁ and V₂, such that every edge connects a vertex in V₁ to a vertex in V₂. No edge connects two vertices within the same set.

Bipartite Graph

If the vertex-set of a graph G can be split into two disjoint sets V₁ and V₂, in such a way that each edge joins a vertex in V₁ to a vertex in V₂, and there are no edges connecting two vertices within V₁ or within V₂, then G is called a bipartite graph.

In the graph above, every edge connects a green vertex (V₁) to a blue vertex (V₂). No edge connects two green vertices or two blue vertices, confirming it is bipartite.

Complete Bipartite Graph

A complete bipartite graph is a bipartite graph in which every vertex in V₁ is connected to every vertex in V₂. It is denoted by K_x,y where x is the number of vertices in V₁ and y is the number of vertices in V₂. The total number of edges in K_x,y is x × y.

In K_3,3 above, every vertex in V₁ is connected to every vertex in V₂, giving 3 × 3 = 9 edges total.

Conclusion

A bipartite graph splits its vertices into two disjoint sets with edges only between the sets. A complete bipartite graph K_x,y connects every vertex in one set to every vertex in the other, resulting in x × y total edges.