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.^[1]  This paper, as well as the one written by Vandermonde on the knight problem, carried on with the analysis situs initiated by Leibniz. Euler's formula relating the  number of edges, vertices, and faces of a convex polyhedron was studied and generalized by Cauchy^[2] and L'Huillier,^[3] and is at the origin of topology.

More than one century after Euler's paper on the bridges of Königsberg and while Listing introduced topology, Cayley was led by the study of particular  analytical forms arising from differential calculus to study a particular class of graphs, the trees.^[4]This study had many implications in theoretical chemistry.

The involved techniques mainly concerned the enumeration of graphs having particular properties. Enumerative graph theory then rose from the results of Cayley and the fundamental results published by Pólya between 1935 and 1937 and the generalization of these by De Bruijn in 1959. Cayley linked his results on trees with the contemporary studies of chemical compositionThe fusion of the ideas coming from mathematics with those coming from chemistry is at the origin of a 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. [...]

I give a rule for the geometrical multiplication of graphs, i.e. for constructing a graph to the product of in- or co-variants whose separate graphs  are given. [...]" (italics as in the original).

The first textbook on graph theory was written by Dénes Kőnig, and published in 1936.^[5] A later textbook by Frank Harary, published in 1969,  was enormously popular, and enabled mathematicians, chemists, electrical engineers and social scientists to talk to each other.

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. Many  incorrect proofs have been proposed, including those by Cayley, Kempe, and others. The study and the generalization of this problem by Tait,  Heawood Ramsey and Hadwiger led to the study of the colorings of the graphs embedded on surfaces with arbitrary genus. Tait's reformulation  generated a new class of problems, the factorization problems, particularly studied by Petersen and Kőnig. 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. In 1969 Heinrich Heesch published a method for solving the problem using  computers.^[11] A computer-aided proof produced in 1976 by Kenneth Appel and Wolfgang Haken makes fundamental use of the notion of "discharging" developed by Heesch. The proof involved checking the properties of 1,936 configurations by computer, and was not fully accepted at the time due to its complexity. 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.

Another important factor of common development of graph theory and topology came from the use of the techniques of modern algebra. The first  example of such a use comes from the work of the physicist Gustav Kirchhoff, who published in 1845 his Kirchhoff's circuit laws for calculating the  voltage and current in electric circuits.

1. Biggs, N.; Lloyd, E. and Wilson, R. (1986), Graph Theory, 1736-1936, Oxford University Press
2.  Cauchy, A.L. (1813), "Recherche sur les polyèdres - premier mémoire", Journal de l'École Polytechnique, 9 (Cahier 16): 66–86.
3. ^ L'Huillier, S.-A.-J. (1861), "Mémoire sur la polyèdrométrie", Annales de Mathématiques 3: 169–189
4. Cayley, A. (1857), "On the theory of the analytical forms called trees", Philosophical Magazine, Series IV 13 (85): 172–176
5. ^ Tutte, W.T. (2001), Graph Theory, Cambridge University Press, p. 30