Can someone ELI5 the real world applications of this? I don't know many people who spend their time coloring graphs, let alone needing to know how many colors are needed to color them.
In compiler technology, "register colouring" is a thing, although no longer necessarily the best technique to solve the particular problem. Optimal Route Computation can be cast as a graph colouring problem, as can scheduling.
I recorded a talk about this some time ago in which I proved that G3C (Graph 3-Colouring) is NPC, let me know if you'd like a link and I'll see if it's still available. If not, I'm giving it again as a part of the G4G Celebration of Mind series of talks.