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Euler's formula states that if a finite, connected, planar graph is drawn in the plane without any edge intersections, and v is the number of vertices, e is the number of edges and f is the number of faces (regions bounded by edges, including the outer, infinitely large region), then + =
This equation, stated by Euler in 1758, [3] is known as Euler's polyhedron formula. [4] ... These transformations eventually reduce the planar graph to a single ...
An Eulerian cycle, [note 1] also called an Eulerian circuit or Euler tour, in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal. [4] The term "Eulerian graph" is also sometimes used in a weaker sense to denote a graph where every vertex has even degree.
Because is a simple planar graph, i.e. it may be embedded in the plane without intersecting edges, and it does not have two vertices sharing more than one edge, and it does not have loops, then it can be shown (using the Euler characteristic of the plane) that it must have a vertex shared by at most five edges. (Note: This is the only place ...
Graph drawing also can be said to encompass problems that deal with the crossing number and its various generalizations. The crossing number of a graph is the minimum number of intersections between edges that a drawing of the graph in the plane must contain. For a planar graph, the crossing number is zero by definition. Drawings on surfaces ...
Kuratowski's theorem states that a finite graph is planar if it is not possible to subdivide the edges of or ,, and then possibly add additional edges and vertices, to form a graph isomorphic to . Equivalently, a finite graph is planar if and only if it does not contain a subgraph that is homeomorphic to K 5 {\displaystyle K_{5}} or K 3 , 3 ...
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 ...
, is a graph with six vertices and nine edges, often referred to as the utility graph in reference to the problem. [1] It has also been called the Thomsen graph after 19th-century chemist Julius Thomsen. It is a well-covered graph, the smallest triangle-free cubic graph, and the smallest non-planar minimally rigid graph.