1-planar graph

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A 1-planar drawing of the Heawood graph: six of the edges have a single crossing, and the remaining 15 edges are not crossed.

In topological graph theory, a 1-planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. If a 1-planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1-plane graph or 1-planar embedding of the graph.

Coloring

1-planar graphs were first studied by Ringel (1965), who showed that they can be colored with at most seven colors.[1] Later, the precise number of colors needed to color these graphs, in the worst case, was shown to be six.[2] The example of the complete graph K6, which is 1-planar, shows that 1-planar graphs may sometimes require six colors. However, the proof that six colors are always enough is more complicated.

Coloring the vertices and faces of the triangular prism graph requires six colors

Ringel's motivation was in trying to solve a variation of total coloring for planar graphs, in which one simultaneously colors the vertices and faces of a planar graph in such a way that no two adjacent vertices have the same color, no two adjacent faces have the same color, and no vertex and face that are adjacent to each other have the same color. This can obviously be done using eight colors by applying the four color theorem to the given graph and its dual graph separately, using two disjoint sets of four colors. However, fewer colors may be obtained by forming an auxiliary graph that has a vertex for each vertex or face of the given planar graph, and in which two auxiliary graph vertices are adjacent whenever they correspond to adjacent features of the given planar graph. A vertex coloring of the auxiliary graph corresponds to a vertex-face coloring of the original planar graph. This auxiliary graph is 1-planar, from which it follows that Ringel's vertex-face coloring problem may also be solved with six colors.[2] The graph K6 cannot be formed as an auxiliary graph in this way, but nevertheless the vertex-face coloring problem also sometimes requires six colors; for instance, if the planar graph to be colored is a triangular prism, then its eleven vertices and faces require six colors, because no three of them may be given a single color.[3]

Edge density

Every 1-planar graph with n vertices has at most 4n − 8 edges.

maximal 1-planar graphs (graphs to which no additional edges can be added while preserving 1-planarity) that have significantly fewer than 4n − 8 edges.[6] The bound of 4n − 8 on the maximum possible number of edges in a 1-planar graph can be used to show that the complete graph K7 on seven vertices is not 1-planar, because this graph has 21 edges and in this case 4n − 8 = 20 < 21.[7]

A 1-planar graph is said to be an optimal 1-planar graph if it has exactly 4n − 8 edges, the maximum possible. In a 1-planar embedding of an optimal 1-planar graph, the uncrossed edges necessarily form a quadrangulation (a

Eulerian (all of its vertices have even degree), that the minimum degree in such a graph is six, and that every optimal 1-planar graph has at least eight vertices of degree exactly six. Additionally, every optimal 1-planar graph is 4-vertex-connected, and every 4-vertex cut in such a graph is a separating cycle in the underlying quadrangulation.[8]

The graphs that have straight 1-planar drawings (that is, drawings in which each edge is represented by a line segment, and in which each line segment is crossed by at most one other edge) have a slightly tighter bound of 4n − 9 on the maximum number of edges, achieved by infinitely many graphs.[9]

Complete multipartite graphs

1-planar drawing of the cocktail party graph K2,2,2,2

A complete classification of the 1-planar

complete multipartite graphs
is known. Every complete bipartite graph of the form K2,n is 1-planar (even planar), as is every complete tripartite graph of the form K1,1,n. Other than these infinite sets of examples, the only complete multipartite 1-planar graphs are K6, K1,1,1,6, K1,1,2,3, K2,2,2,2, K1,1,1,2,2, and their subgraphs. The minimal non-1-planar complete multipartite graphs are K3,7, K4,5, K1,3,4, K2,3,3, and K1,1,1,1,3. For instance, the complete bipartite graph K3,6 is 1-planar because it is a subgraph of K1,1,1,6, but K3,7 is not 1-planar.[7]

Computational complexity

It is

polynomial time when those parameters are bounded.[13]

In contrast to

book thickness,[18] but some 1-planar graphs including K2,2,2,2 have book thickness at least four.[19]

1-planar graphs have

maximum independent set of a 1-planar graph.[20]

Generalizations and related concepts

The class of graphs analogous to

linear time.[22][23] The triconnected components of the graph (nodes of the SPQR tree) can consist only of cycle graphs, bond graphs, and four-vertex complete graphs, from which it also follows that outer-1-planar graphs are planar and have treewidth
at most three.

The 1-planar graphs include the 4-map graphs, graphs formed from the adjacencies of regions in the plane with at most four regions meeting in any point. Conversely, every optimal 1-planar graph is a 4-map graph. However, 1-planar graphs that are not optimal 1-planar may not be map graphs.[24]

1-planar graphs have been generalized to k-planar graphs, graphs for which each edge is crossed at most k times (0-planar graphs are exactly the planar graphs). Ringel defined the local crossing number of G to be the least non-negative integer k such that G has a k-planar drawing. Because the local crossing number is the maximum

sparse graphs, implying that the 1-planar and k-planar graphs have bounded expansion.[28]

Nonplanar graphs may also be parameterized by their crossing number, the minimum number of pairs of edges that cross in any drawing of the graph. A graph with crossing number k is necessarily k-planar, but not necessarily vice versa. For instance, the Heawood graph has crossing number 3, but it is not necessary for its three crossings to all occur on the same edge of the graph, so it is 1-planar, and can in fact be drawn in a way that simultaneously optimizes the total number of crossings and the crossings per edge.

Another related concept for nonplanar graphs is

graph skewness
, the minimal number of edges that must be removed to make a graph planar.

References

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Further reading
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