> graph coloring register allocation
I knew some people working on linear optimization, at the time one of the most performance-intensive applications around (with real money/competitiveness in many industries riding on it). The compiler that produced the best code, by quite a bit, was the IBM FORTRAN compiler (3090 was the preferred target at the time), which also didn't do graph coloring. It just allocated the registers by loops from innermost to outermost.
¯\_(ツ)_/¯\
For a simple compiler, we should also look at Wirth's works, and Wirth's rule: an optimization has to pay for itself, that is, the compiler compiling itself with the optimization both in the executable and in the source code has to be faster than without it in both places.