Is there a simple proof for the 4-color theorem when restricted to (finite) maps all of whose regions are pentagons? I am in fact most interested in convex pentagons, if that additional structure helps. These maps lead to graphs of maximum degree $\Delta=5$, so a result for such graphs will answer my question.
If all regions are quadrilaterals, then four colors are sometimes necessary,
even if all quadrilaterals are convex, e.g.,
But here there is a simple proof of 4-colorability: Identify a quadrilateral with an exposed edge, remove it, 4-color the remainder by induction, and replace the quad, coloring it with a color different from its at most three neighbors. Obviously this simple proof fails for pentagons.
Especially the $\Delta=5$ case is likely known to the experts. So a reference would suffice. Thanks!