A generalization of Euler's formula for arbitrary planar graphs exists: F - E + V - C = 1, where C is the number of components in the graph. In 1736 Euler solved, or rather proved insoluble, a problem known as the seven bridges of Königsberg, publishing a paper Solutio ...
Euler's formula exhibits a beautiful relation between the number of vertices, edges and faces that is valid for any plane graph. Euler mentioned this result for the first time in a letter to his friend Goldbach in 1750, but he did not have a complete proof at the time. Among the many ...
A graph isplanarif it can be drawn in the plane ℝ2without crossing edges (or, equivalently, on the 2-dimensional sphereS2). We talk of aplanegraph if such a drawing is already given and fixed. Any such drawing decomposes the plane or sphere into a finite number of connected regions,...
We will refer to a planar map as connected if the graph that gives rise to it is connected. Figure 8.18b is not a connected map (it has three components), whereas the maps in Figures 8.18a, c, and d are connected. For both the proof and some applications of Euler's formula, we ...
When f is a stationary magnetic field, so that divf=0 and curlf= current density, by Maxwell’s equations, the above formula agrees with Biot – Savart’s law found in textbooks. If f instead represents a stationary electric field, then curlf=0 and we obtain Coulomb’s law for an ...
Then, we can calculate the number of connected components and holes by counting the different labels, and thus, Formula (1) can be used for calculating the Euler number. In Ref. [9], He et al. presented an efficient labeling algorithm for distinguishing the different connected components and...