Colorings

This chapter treats a subject occurring quite often in graph theory: colorings. We shall prove the two fundamental major results in this area, namely the theorems of Brooks on vertex colorings and the theorem of Vizing on edge colorings. As an aside, we explain the relationship between colorings and partial orderings, and briefly discuss perfect graphs. Moreover, we consider edge colorings of Cayley graphs; these are graphs which are defined using groups. Finally, we turn to map colorings: we shall prove Heawood’s five color theorem and report on the famous four color theorem. Our discussion barely scratches the surface of the vast area; for a detailed study of coloring problems we refer the reader to the monograph [JeTo95].