Every complete graph K has treewidth n 1.

The example of the complete graph K, which is 1-planar, shows that 1-planar graphs may sometimes require six colors.

The complete bipartite graph K has edge covering number max(m, n).

The complete graph K is integral for all n.

The graph K embeds on every surface except for the sphere.

The graph K, for example, has 6 vertices, 9 edges, and no cycles of length 3.

The complete graph K is also in the Petersen family.

That is, by assuming a higher level of connectivity, the other graph K can be made unnecessary in the characterization.

The graph K is called invariant or sometimes the gluing graph.

A complete bipartite graph K has m n spanning trees.