The puzzle involved finding a special type of path, later known as a Hamiltonian circuit , along the edges of a dodecahedron a Platonic solid consisting of 12 pentagonal faces that begins and ends at the same corner while passing through each corner exactly once. Hamiltonian graphs have been more challenging to characterize than Eulerian graphs, since the necessary and sufficient conditions for the existence of a Hamiltonian circuit in a connected graph are still unknown.
The histories of graph theory and topology are closely related, and the two areas share many common problems and techniques. The vertices and edges of a polyhedron form a graph on its surface, and this notion led to consideration of graphs on other surfaces such as a torus the surface of a solid doughnut and how they divide the surface into disklike faces.
Having considered a surface divided into polygons by an embedded graph, mathematicians began to study ways of constructing surfaces, and later more general spaces, by pasting polygons together. The connection between graph theory and topology led to a subfield called topological graph theory. An important problem in this area concerns planar graphs.
Encyclopaedia of Mathematical Sciences Low Dimensional Topology. Free Preview cover. © Graphs on Surfaces and Their Applications. Authors: Lando. Graphs drawn on two-dimensional surfaces have always attracted Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume ).
These are graphs that can be drawn as dot-and-line diagrams on a plane or, equivalently, on a sphere without any edges crossing except at the vertices where they meet. Complete graphs with four or fewer vertices are planar, but complete graphs with five vertices K 5 or more are not. Nonplanar graphs cannot be drawn on a plane or on the surface of a sphere without edges intersecting each other between the vertices.
The use of diagrams of dots and lines to represent graphs actually grew out of 19th-century chemistry , where lettered vertices denoted individual atoms and connecting lines denoted chemical bonds with degree corresponding to valence , in which planarity had important chemical consequences. The first use, in this context , of the word graph is attributed to the 19th-century Englishman James Sylvester , one of several mathematicians interested in counting special types of diagrams representing molecules. Another class of graphs is the collection of the complete bipartite graphs K m , n , which consist of the simple graphs that can be partitioned into two independent sets of m and n vertices such that there are no edges between vertices within each set and every vertex in one set is connected by an edge to every vertex in the other set.
In the Polish mathematician Kazimierz Kuratowski proved that any nonplanar graph must contain a certain type of copy of K 5 or K 3, 3. While K 5 and K 3, 3 cannot be embedded in a sphere, they can be embedded in a torus. The graph-embedding problem concerns the determination of surfaces in which a graph can be embedded and thereby generalizes the planarity problem. It was not until the late s that the embedding problem for the complete graphs K n was solved for all n. Another problem of topological graph theory is the map-colouring problem.
Of central interest is the question how the invariant ring of a given group action can be calculated. Buldygin; Yu. Zhuravlev, D. When each vertex is connected by an edge to every other vertex, the graph is called a complete graph. Waldspurger; C. Vassiliev Editor. Recent Developments in Several Complex Variables.
This problem is an outgrowth of the well-known four-colour map problem , which asks whether the countries on every map can be coloured by using just four colours in such a way that countries sharing an edge have different colours. Asked originally in the s by Francis Guthrie, then a student at University College London, this problem has a rich history filled with incorrect attempts at its solution.
In an equivalent graph-theoretic form, one may translate this problem to ask whether the vertices of a planar graph can always be coloured by using just four colours in such a way that vertices joined by an edge have different colours. The result was finally proved in by using computerized checking of nearly 2, special configurations.
Interestingly, the corresponding colouring problem concerning the number of colours required to colour maps on surfaces of higher genus was completely solved a few years earlier; for example, maps on a torus may require as many as seven colours. This work confirmed that a formula of the English mathematician Percy Heawood from correctly gives these colouring numbers for all surfaces except the one-sided surface known as the Klein bottle , for which the correct colouring number had been determined in Among the current interests in graph theory are problems concerning efficient algorithms for finding optimal paths depending on different criteria in graphs.
Two well-known examples are the Chinese postman problem the shortest path that visits each edge at least once , which was solved in the s, and the traveling salesman problem the shortest path that begins and ends at the same vertex and visits each edge exactly once , which continues to attract the attention of many researchers because of its applications in routing data, products, and people. Work on such problems is related to the field of linear programming , which was founded in the midth century by the American mathematician George Dantzig.
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Thank you for your feedback. Written By: Stephan C. See Article History. The ideas are mostly quite elementary and accessible, and one can see how they might work well in a math circle for bright and interested high school students. Each of the following chapters takes those ideas and develops them in the context of different areas of mathematics.
The matrix integrals method comes from theoretical physics and has to do with probability distributions on spaces of matrices. As one might expect, none of these chapters offers a complete treatment of their subject, but they are surprisingly readable and often quite interesting. Though many details are omitted, readers who have a good grasp of the material in the first chapter and who are willing to suspend disbelief as necessary can probably get quite a bit out of them.
Don Zagier contributes a delightful appendix on the representation theory of finite groups, in which all of the elementary theory is developed in less than ten pages. It left me dreaming of a seminar for advanced undergraduates in which one would hand out those pages and then spend a semester unpacking them. I wonder if it would work…. Never mind: the goal of the examples given below is to show that you may get some useful information even without a profound knowledge of the subject.
We had a chance to give the explanations presented below at a number of seminars in the presence of physicists. Usually our statements did not produce any serious protests. Hoping that the politeness of our colleagues-physicists was not the unique reason for that…. As that sentence also shows, a few Slavicisms have made it into the text, but they are never annoying. Every so often there is a hint of truly strange things:.
An even more complicated class of models considers the surfaces not as abstract entities but as geometric objects embedded or immersed in a certain space. See in this respect , where embeddings of surfaces into a space of dimension —2 are considered! Indeed, the level of abstraction in modern physics is often far beyond that of mathematics.
The overall effect is to draw the reader in and to open up a wide vista of little-known mathematics that is full of surprising ideas, unexpected connections, and just-plain-wonderful theorems. This is a tour worth taking.
Fernando Q. He lives in a space whose dimension is definitely positive. Skip to main content. Search form Search.