For a planar graph, the crossing number is zero by definition.
F is a minor-closed graph family that does not include all planar graphs.
Euler also made contributions to the understanding of planar graphs.
All planar graphs have book thickness at most four.
It was claimed in that some planar graphs have book thickness exactly four.
There exist planar non-Hamiltonian graphs in which all faces have five or eight sides.
By 1967, Kasteleyn had generalized this result to all planar graphs.
View the diagram (the circle together with all the chords) above as a planar graph.
These transformations eventually reduce the planar graph to a single triangle.
Their proof also leads to an efficient algorithm for finding 4-colorings of planar graphs.
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