Definition:Complete Bipartite Graph

A complete bipartite graph is a bipartite graph $$G = \left({A | B, E}\right)$$ in which every vertex in $$A$$ is adjacent to every vertex in $$B$$.

The complete bipartite graph where $$A$$ has $$m$$ vertices and $$B$$ has $$n$$ vertices is denoted $$K_{m, n}$$.

Note that $$K_{m, n}$$ is the same as $$K_{n, m}$$.

Examples

 * CompleteBipartiteGraphs.png

Basic Properties

 * $$K_{1, n}$$ = $$K_{n, 1}$$ is a tree for all $$n$$, and no other complete bipartite graphs are trees.


 * $$K_{n, n}$$ is $n$-regular for all $$n$$, and no other complete bipartite graphs are regular.


 * $$K_{n, n}$$ is Hamiltonian for all $$n \ge 2$$, from Complete Hamiltonian Bipartite Graph, and no other complete bipartite graphs are Hamiltonian.


 * $$K_{1, 1}$$ semi-Hamiltonian, as is $$K_{n, n+1}$$ for all $$n \ge 2$$, from Condition for Complete Bipartite Graph to be Semi-Hamiltonian, and no other complete bipartite graphs are semi-Hamiltonian.


 * $$K_{1, 1}$$ is the complete graph $$K_2$$, and no other complete bipartite graphs are complete.


 * $$K_{1, 1}$$ and $$K_{1, 2}$$ are the path graphs $$P_2$$ and $$P_3$$ and no other complete bipartite graphs are path graphs.


 * $$K_{2, 2}$$ is the cycle graph $$C_4$$, and no other complete bipartite graphs are cycle graphs.


 * $$K_{1, 3}$$ is known as a claw.