Complement of Complete Bipartite Graph

Theorem
Let $K_{p, q}$ be a complete bipartite graph.

The complement of $K_{p, q}$ consists of a disconnected graph with two components:


 * The complete graph $K_p$


 * The complete graph $K_q$.

Proof
By definition, the complete bipartite graph $K_{p, q}$ consists of two sets of vertices: $A$ of cardinality $p$, and $B$ of cardinality $q$, such that:


 * Every vertex in $A$ is adjacent to every vertex in $B$


 * No vertex in $A$ is adjacent to any other vertex in $A$


 * No vertex in $B$ is adjacent to any other vertex in $B$.

The complement of $K_{p, q}$ therefore must be a graph $G$ such that:


 * No vertex in $A$ is adjacent to any vertex in $B$


 * Every vertex in $A$ is adjacent to every other vertex in $A$


 * Every vertex in $B$ is adjacent to every other vertex in $B$.

The second and third of these conditions describes the complete graphs $K_p$ and $K_q$.

From the first of these conditions, it follows that $G$ comes in two disconnected graph components.

Hence the result.