Graph Coloring: The Four Color Problem

Graph coloring has been one of the most popular areas of graph theory and is a topic that relates to a variety of real-life examples. The origin of graph coloring can be traced back to the famous \textit{Four Color Problem},~\cite{gary08} posed more than 150 years ago, relating to the coloring of maps.

It all started in the year 1952 when a British mathematician named Francis Guthrie, while coloring the countries of a map of England, observed that he could color them with four colors, and later conjectured that four colors would be sufficient to color the regions of any map. By observation, it was found that the conjecture was true but proving it remained an open problem for a considerable period of time.

{\bf The Four Color Problem} \textit{Is it possible to color the countries of every map with four or fewer colors so that every two countries sharing a common boundary are colored differently?}

Even then, Kempe's paper helped other mathematicians to attack this problem using various approaches. Finally, after more than a century passed since the problem came into existence, in the year 1976, two mathematicians Wolfgang Haken and Kenneth Appel~\cite{kapp76}, taking the help of a computer, proved that the conjecture was indeed true. The proof consisted of a rather short theoretical section, four hundred pages of detailed checklists showing that all relevant cases had been covered, and about 1800 computer runs totaling over a thousand hours of computer time. However, not many accepted this proof partly because of the extensive use of computers in it and partly because of its extraordinary length. After this, several versions of the proof of the Four Color Theorem came into existence out of which the most acceptable and well-structured was given by Neil Robertson, Daniel P. Sanders, Paul Seymour and Robin Thomas~\cite{nrob96} in the year