Degree Bounds for the Chromatic Number

Abstract

Until the 1940s, coloring theory focused almost exclusively on coloring of maps. However, already in 1879 A. B. Kempe suggested coloring of abstract graphs as a possible topic. Around 1930 H. A. Whitney considered chromatic polynomials of graphs, rather than maps, in his Harvard thesis and in the resulting paper of 1932 about coloring of graphs. But it was only with the seminal paper by R. L. Brooks in 1941 that coloring of abstract graphs emerged as a topic of study in its own right. Brooks’ result, which has become known as Brooks’ theorem, relates the chromatic number to the maximum degree of a given graph. Over the years, graph coloring theory has developed into a rich theory and, as emphasized by B. Reed in his extensive paper in 1998 about 𝜔, Δ, and 𝜒, Brooks’ theorem is just the tip of the iceberg.