Easy instances of the coloring problem on graphs with degree at most 4 Given a graph with some set of colors, the goal of the coloring problem is to color the input graph with as few number of colors as possible, such that no adjacent vertices have the same color.

In general, the problem is hard. For trees and cycles, the problem is solvable in polynomial time. I am interested in bounded degree instances or more precisely graphs of degree at most 4. Is there any non-trivial subclass of graphs of degree at most 4 for which the coloring problem is solvable in polynomial time?
You can of course take any graph class for which coloring is easy, and additionally require that the maximum degree is at most 4. For example, every bipartite graph of maximum degree at most 4 works. Or "bipartite" could be replaced with say "outerplanar" or more generally "small cliquewidth".

But having maximum degree at most 4 alone will not work. That is, coloring remains hard for (i) 4-regular graphs and (ii) planar graphs of maximum degree 4.