Degeneracy and Colorings

As discussed in the previous chapter, every graph 𝐺 satisfies 𝜒(𝐺) ≤ col(𝐺) ≤ Δ(𝐺) + 1. However, for many graph classes, the difference between the coloring number and the maximum degree can be arbitrarily large. For example, planar graphs have unbounded maximum degree, but their coloring number is at most 6. While Brooks’ theorem provides a characterization of graphs satisfying 𝜒 = Δ+ 1, a characterization of graphs satisfying 𝜒 = col seems to be unattainable. A graph satisfies col = Δ+ 1 if and only if it has a Δ-regular component.