stylistic:tone
Other results

The four color theorem states that every planar graph is 4- colorable (i.e., 4-partite).

Fáry's theorem states that every simple planar graph admits a representation as a planar straight-line graph. A universal point set is a set of points such that every planar graph with n vertices has such an embedding with all vertices in the point set; there exist universal point sets of quadratic size, formed by taking a rectangular subset of the integer lattice. Every simple outerplanar graph admits an embedding in the plane such that all vertices lie on a fixed circle and all edges are straight line segments that lie inside the disk and don't intersect, so n -vertex regular polygons are universal for outerplanar graphs.

Scheinerman's conjecture (now a theorem) states that every planar graph can be represented as an intersection graph of line segments in the plane.

The planar separator theorem states that every n -vertex planar graph can be partitioned into two subgraphs of size at most 2 n /3 by the removal of O(√ n) vertices. As a consequence, planar graphs also have treewidth and branch-width O(√ n).

The planar product structure theorem states that every planar graph is a subgraph of the strong graph product of a graph of treewidth at most 8 and a path. This result has been used to show that planar graphs have bounded queue number, bounded non-repetitive chromatic number, and universal graphs of near-linear size. It also has applications to vertex ranking and p -centered colouring of planar graphs.

For two planar graphs with v vertices, it is possible to determine in time O(v) whether they are isomorphic or not (see also graph isomorphism problem).

Any planar graph on n nodes has at most 8(n-2) maximal cliques, which implies that the class of planar graphs is a class with few cliques.