Such weighted graphs are commonly used to program GPS's, and travel-planning search engines that compare flight times and costs. == History == The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.

Harary donated all of the royalties to fund the Pólya Prize. One of the most famous and stimulating problems in graph theory is the four color problem: "Is it true that any map drawn in the plane may have its regions colored with four colors, in such a way that any two regions having a common border have different colors?" This problem was first posed by Francis Guthrie in 1852 and its first written record is in a letter of De Morgan addressed to Hamilton the same year.

A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas. The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney.

The fusion of ideas from mathematics with those from chemistry began what has become part of the standard terminology of graph theory. In particular, the term "graph" was introduced by Sylvester in a paper published in 1878 in Nature, where he draws an analogy between "quantic invariants" and "co-variants" of algebra and molecular diagrams: "[…] Every invariant and co-variant thus becomes expressible by a graph precisely identical with a Kekuléan diagram or chemicograph.

A simpler proof considering only 633 configurations was given twenty years later by Robertson, Seymour, Sanders and Thomas. The autonomous development of topology from 1860 and 1930 fertilized graph theory back through the works of Jordan, Kuratowski and Whitney.

Enumerative graph theory then arose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937.

[…]" (italics as in the original). The first textbook on graph theory was written by Dénes Kőnig, and published in 1936.

Enumerative graph theory then arose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937.

The works of Ramsey on colorations and more specially the results obtained by Turán in 1941 was at the origin of another branch of graph theory, extremal graph theory. The four color problem remained unsolved for more than a century.

These were generalized by De Bruijn in 1959.

Another book by Frank Harary, published in 1969, was "considered the world over to be the definitive textbook on the subject", and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other.

In 1969 Heinrich Heesch published a method for solving the problem using computers.

A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of "discharging" developed by Heesch.

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