"More precisely, it is NP-hard1. So it is unlikely that we will find an efficient algorithm to optimally solve the coloring problem on an arbitrary input graph. This fundamental hardness result casts a long shadow across the landscape of graph coloring."
Yes, but in practice heuristic approaches can work well, even if there are no guarantees of optimality. See for example my response about a simple "most constrained first" heuristic.
