Comment by erikvanoosten
13 hours ago
You can convert edge coloring problems into vertex coloring problems and vice versa through a simple O(n) procedure.
13 hours ago
You can convert edge coloring problems into vertex coloring problems and vice versa through a simple O(n) procedure.
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.
For the edge-coloring problem, the optimal number of colors needed to properly color the edges of G is always either Delta(G) (the maximum degree of G) or Delta(G) + 1, but deciding which one is the true optimum is an NP-complete problem.
Nevertheless, you can always properly edge-color a graph with Delta(G) + 1 colors. Finding such a coloring could in principle be slow, though: the original proof that Delta(G) + 1 colors is always doable amounted to a O(e(G) * v(G)) algorithm, where e(G) and v(G) denote the number of edges and vertices of G, respectively. This is polynomial, but nowhere near linear. What the paper in question shows is how, given any graph G, to find an edge coloring using Delta(G) + 1 colors in O(e(G) * log(Delta(G))) time, which is linear time if the maximum degree is a constant.
Yes. The article ran through this point as follows:
"In 1964, a mathematician named Vadim Vizing proved a shocking result: No matter how large a graph is, it’s easy to figure out how many colors you’ll need to color it. Simply look for the maximum number of lines (or edges) connected to a single point (or vertex), and add 1."
I keep wondering why I ever read Quanta Magazine. It takes a pretty generous reading of "need" to make this a correct statement.