By the four color theorem, every planar graph can be 4-colored.

For a planar graph, the crossing number is zero by definition.

It was claimed in that some planar graphs have book thickness exactly four.

Conversely any planar graph can be formed from a map in this way.

These transformations eventually reduce the planar graph to a single triangle.

Every planar graph whose faces all have even length is bipartite.

Euler also made contributions to the understanding of planar graphs.

All planar graphs have book thickness at most four.

View the diagram (the circle together with all the chords) above as a planar graph.

By 1967, Kasteleyn had generalized this result to all planar graphs.

This article is about the game; for the graph theory property, see Planar graph.

The Euler characteristic of any planar graph is 2 .

Also, a tree is necessarily a planar graph.

Their proof also leads to an efficient algorithm for finding 4-colorings of planar graphs.

The Hosoya index is to compute, even for planar graphs.

F is a minor-closed graph family that does not include all planar graphs.

The 3-coloring problem remains NP-complete even on planar graphs of degree 4.

A slightly more general result is true: if a planar graph has at most three triangles then it is 3-colorable.

Therefore, by Steinitz's theorem, it is a 3-vertex-connected simple planar graph.

If this were the restriction, planar graphs would require arbitrarily large numbers of colors.

In 1981, he solved Ulam's problem, showing that there exists no universal planar graph.

Therefore, Apollonian networks may also be characterized as the uniquely 4-colorable planar graphs.

In particular, every planar graph has a planar arc diagram.

However, every planar graph has an arc diagram in which each edge is drawn with at most two semicircles.

The intuitive idea underlying discharging is to consider the planar graph as an electrical network.