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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. [16] 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.
The Euler characteristic can be defined for connected plane graphs by the same + formula as for polyhedral surfaces, where F is the number of faces in the graph, including the exterior face. The Euler characteristic of any plane connected graph G is 2.
This is known as Euler's Theorem: A connected graph has an Euler cycle if and only if every vertex has an even number of incident edges. The term Eulerian graph has two common meanings in graph theory. One meaning is a graph with an Eulerian circuit, and the other is a graph with every vertex of even degree.
For planar graphs, the properties of being Eulerian and bipartite are dual: a planar graph is Eulerian if and only if its dual graph is bipartite. As Welsh showed, this duality extends to binary matroids: a binary matroid is Eulerian if and only if its dual matroid is a bipartite matroid, a matroid in which every circuit has even cardinality.
By Euler's formula for planar graphs, there are exactly + bounded faces. The symmetric difference of any set of face cycles is the boundary of the corresponding set of faces, and different sets of bounded faces have different boundaries, so it is not possible to represent the same set as a symmetric difference of face cycles in more than one ...
Euler also discovered the formula + = relating the number of vertices, edges, and faces of a convex polyhedron, [92] and hence of a planar graph. The constant in this formula is now known as the Euler characteristic for the graph (or other mathematical object), and is related to the genus of the object. [93]
Euler also made contributions to the understanding of planar graphs. He introduced a formula governing the relationship between the number of edges, vertices, and faces of a convex polyhedron. Given such a polyhedron, the alternating sum of vertices, edges and faces equals a constant: V − E + F = 2.
Next follow chapters on spanning trees and Cayley's formula, chemical graph theory and graph enumeration, and planar graphs, Kuratowski's theorem, and Euler's polyhedral formula. There are three chapters on the four color theorem and graph coloring, a chapter on algebraic graph theory, and a final chapter on graph factorization. Appendices ...