The Fastest Way yet to Color Graphs (opens in new tab)

Wrong. You can convert edge-coloring problems into vertex-coloring problems of the so-called line graph: https://en.m.wikipedia.org/wiki/Line_graph

But the opposite is not true, because not every graph is a line graph of some other graph.

Hrm... right. It's been a while. And it looks like both Vertex Coloring and Edge Coloring are both NP-complete (because of the O(n) procedure you're talking about and the ability to reduce both problems down to 3-SAT). I've started looking closer at the actual paper to try to figure out what's going on here. Thanks for the reminder, I miss getting to regularly work on this stuff.

Edit: thanks sibling reply for pointing out that it's not a bidirectional transform.

Totally. The hard part isn't coloring (you can use simple heuristics to get a decent register assignment), rather, it's figuring out which registers to spill (don't spill registers in hot loops! and a million other things!).

and this post isn't even about vertex coloring