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A planar graph is said to be convex if all of its faces (including the outer face) are convex polygons. Not all planar graphs have a convex embedding (e.g. the complete bipartite graph K 2,4). A sufficient condition that a graph can be drawn convexly is that it is a subdivision of a 3-vertex-connected planar graph.
This equation, stated by Euler in 1758, [2] is known as Euler's polyhedron formula. [3] ... These transformations eventually reduce the planar graph to a single ...
The two graphs and , are nonplanar, as may be shown either by a case analysis or an argument involving Euler's formula. Additionally, subdividing a graph cannot turn a nonplanar graph into a planar graph: if a subdivision of a graph has a planar drawing, the paths of the subdivision form curves that may be used to represent the edges of itself ...
K 1 through K 4 are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph K 5 plays a key role in the characterizations of planar graphs: by Kuratowski's theorem, a graph is planar if and only if it contains neither K 5 nor the complete bipartite ...
The proof is an easy consequence of Euler's formula. [1] [2] As a corollary of this theorem, if an embedded planar graph has only one face whose number of sides is not 2 mod 3, and the remaining faces all have numbers of sides that are 2 mod 3, then the graph is not Hamiltonian.
In graph theory, Mac Lane's planarity criterion is a characterisation of planar graphs in terms of their cycle spaces, named after Saunders Mac Lane who published it in 1937. It states that a finite undirected graph is planar if and only if the cycle space of the graph (taken modulo 2) has a cycle basis in which each edge of the graph ...
In graph-theoretic terms, the theorem states that for loopless planar graph, its chromatic number is ().. The intuitive statement of the four color theorem – "given any separation of a plane into contiguous regions, the regions can be colored using at most four colors so that no two adjacent regions have the same color" – needs to be interpreted appropriately to be correct.
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 ...