In SSA, the graphs are chordal, so were already easily colorable (relatively).
Outside of SSA, this is not true, but the coloring is still not the hard part, it's the easy part.
If you read the introduction, you'll also read that the goal is to "color each of your lines and require that for every point, no two lines connected to it have the same color."
Ps. "How many colors a graph needs" is a very well established term in computer science and graph theory.
Personally, I would interpret this as option 1 (and so did the comment above I assume). In that case, the statement is wrong. But I’d prefer to specify „at most/ at least“ anyways.
Or even better, use actual vocabulary. „For every graph there exists a coloring with X colors.“ or „any graph can be coloured using X colors“.
PS: I also agree with the sentiment about quanta magazine. It’s hard to get some actual information from their articles if you know the topic.
No matter how large a car is, it is easy to figure out how much money you'll need to buy it. Simply look at the price tag.
(From: 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 ...)
tonyarkles•1y ago
erikvanoosten•1y ago
meindnoch•1y ago
But the opposite is not true, because not every graph is a line graph of some other graph.
erikvanoosten•1y ago
tonyarkles•1y ago
Edit: thanks sibling reply for pointing out that it's not a bidirectional transform.
mauricioc•1y ago
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.
Syzygies•1y ago
"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.
JohnKemeny•1y ago