ABSTRACT
This chapter describes the coloring of a graph and its chromatic number, the stage is set for a proof of the Five Color Theorem and discusses the Four Color Conjecture. It explores uniquely colorable graphs, which can only be colored in one way, and critical graphs, which are minimal with respect to coloring. The chapter explains the intimate relationship between homomorphisms and colorings. A coloring of a graph is an assignment of colors to its points so that no two adjacent points have the same color. The chapter shows that one may very well be led to believe that all graphs with large chromatic number have large cliques and hence contain triangles. Although it is not known whether all planar graphs are 4-colorable, they are certainly 5-colorable. The chapter concludes with a development of the properties of the chromatic polynomial.
