HISTORY
The origin of graph theory can be traced back to Euler's work on the Konigsberg bridges problem (1735), which subsequently led to the concept of an Eulerian graph. The study of cycles on polyhedra by the Thomas P. Kirkman (1806 - 95) and William R. Hamilton (1805-65) led to the concept of a Hamiltonian graph.
The concept of a tree, a connected graph without cycles, appeared implicitly in the work of Gustav Kirchhoff (1824-87), who employed graph-theoretical ideas in the calculation of currents in electrical networks or circuits. Later, Arthur Cayley (1821-95), James J. Sylvester(1806-97), George Polya(1887-1985), and others use 'tree' to enumerate chemical molecules.
The study of planar graphs originated in two recreational problems involving the complete graph K5 and the complete bipartite graph K3,3. These graphs proved to be planarity, as was subsequently demonstrated by Kuratowski. First problem was presented by A. F. Mobius around the year 1840 as follows
Once upon a time, there was a king with five sons.
In his will he stated that after his death the sons
should divide the kingdom into five provinces so
that the boundary of each province should have
a frontiers line in common with each of the other
four provinces.
Here the problem is whether one can draw five mutually neighboring regions in the plane.
The king further stated that all five brothers should
join the provincial capital by roads so that no two
roads intersect.
Here the problem is that deciding whether the graph K5 is planar.
The origin of second problem is unknown but it is first mentioned by H. Dudeney in 1913 in its present form.
The puzzle is to lay a water, gas, and electricity
to each of the three houses without any pipe
crossing another.
This problem is that of deciding whether the graph K3,3 is planar.
The celebrated four-color problem was first posed by Francis Guthrie in 1852. and a celebrated incorrect "proof" by appeared in 1879 by Alfred B. Kempe. It was proved by Kenneth Appel and Wolfgang Haken in 1976 and a simpler and more systematic proof was produced by Neil Roberton, Daniel Sanders, Paul Seymour, and Robin Thomas in 1994.