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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.
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.
[8] 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.
Map of Königsberg in Euler's time showing the actual layout of the seven bridges, highlighting the river Pregel and the bridges. The Seven Bridges of Königsberg is a historically notable problem in mathematics. Its negative resolution by Leonhard Euler, in 1736, [1] laid the foundations of graph theory and prefigured the idea of topology. [2]
One direction of the characterisation states that every planar graph has a 2-basis. Such a basis may be found as the collection of boundaries of the bounded faces of a planar embedding of the given graph G. If an edge is a bridge of G, it appears twice on a single face boundary and therefore has a zero coordinate in the corresponding vector ...
In 1736, Euler showed that G has an Eulerian circuit if and only if G is connected and the indegree is equal to outdegree at every vertex. In this case G is called Eulerian. We denote the indegree of a vertex v by deg(v). The BEST theorem states that the number ec(G) of Eulerian circuits in a connected Eulerian graph G is given by the formula
Euler's theorem, on modular exponentiation; Euler's partition theorem relating the product and series representations of the Euler function Π(1 − x n) Goldbach–Euler theorem, stating that sum of 1/(k − 1), where k ranges over positive integers of the form m n for m ≥ 2 and n ≥ 2, equals 1; Gram–Euler theorem
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