I hope you enjoyed this peek behind the curtain at how graph theory the math that powers graph technology looks at the world through an entirely different lens that solves problems in new and. Graph theory and cayleys formula university of chicago. The induction is obvious for m0 since in this case n1 and f1. This is my favorite proof, and is the one i use when teaching graph theory. Graph theory has experienced a tremendous growth during the 20th century. The more wellknown 4coloring theorem is much harder to prove. Even after a course on graph theory, i did not realise it. We dont talk about faces of a graph unless the graph is drawn without any overlaps.
Just before i tell you what euler s formula is, i need to tell you what a face of a plane graph is. Despite its seemingly narrow formulation, the marriage theorem. In recent years, graph theory has established itself as an important mathematical. In mathematics, graph theory is the study of graphs, which are mathematical structures used to.
In this video, 3blue1brown gives a description of planar graph duality and how it can be applied to a proof of euler s characteristic formula. Eulers formula relating the number of edges, vertices, and faces of a convex polyhedron was studied and generalized by cauchy and. Euler s formula proof using mathematical induction method graph theory lectures duration. A face is a region between edges of a plane graph that doesnt have any edges in it. A graph is polygonal is it is planar, connected, and has the property that every edge borders on two different faces.
We can prove the formula for all connected planar graphs, by induction on the number. This book aims to provide a solid background in the basic topics of graph theory. Cayleys formula is one of the most simple and elegant results in graph theory, and as a result, it lends itself to many beautiful proofs. As part of my cs curriculum next year, there will be some graph theory involved and this book covers much much more and its a perfect introduction to the subject. A description of planar graph duality, and how it can be applied in a particularly elegant proof of eulers characteristic formula. In this lecture we are going to learn about euler s formula and we proof that formula by using mathematical induction euler s formula in graph theory proof of euler s formula by using mathematical.
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