A 1-planar graph is a graph that can be drawn in the plane such that each edge has at most one crossing point at which it crosses a single additional edge (Ringel 1965).
A 1-planar graph with 
vertices has at most 
edges. Such optimal 1-planar graphs have been completely characterized.
4-map graphs are 1-planar.
Fabrici and Madaras (2007) showed that that a 1-planar graph has minimum vertex degree 
and that every 3-connected 1-planar graph contains an edge with both endvertices
of degrees at most 20.
Borodin (1984) proved that 1-planar graphs are 6-colorable (Fabrici and Madara 2007).
A 1-planar drawing has at most 
crossings.
See also
2-Planar Graph,
Singlecross
Graph
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References
Borodin, O. V. "Solution of Ringel's Problems on the Vertex-Face Coloring of Plane Graphs and on the Coloring of 1-Planar Graphs."
Diskret. Anal. Novosibirsk 41, 12-26, 1984.Brandenburg,
F. J. "Straight-Line Drawings of 1-Planar Graphs." 3 Sep 2021. https://arxiv.org/abs/2109.01692.Fabrici,
I. and Madaras, T. "The Structure of 1-Planar Graphs." Disc. Math. 307,
854-865, 2007.Ringel, G. "Ein Sechsfarbenproblem auf der Kugel."
Abh. Math. Sem. Univ. Hamburg 29, 107-117, 1965.
Cite this as:
Weisstein, Eric W. "1-Planar Graph." From
MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/1-PlanarGraph.html
