Let g be planar when is the upper bound on g better from

Question

Question: 1. Let G be planar. When is the upper bound on χ(G) better from planarity than from Δ(G)+1?

2. Use Theorem to determine χ' (K_m,n).

Theorem: For bipartite G, χ' (G) = Δ(G)

3. Challenge: Create a graph on which a greedy algorithm produces a truly awful vertex coloring.