TITLE: Why is edge-coloring less interesting than vertex-coloring?
QUESTION [21 upvotes]: I was wondering why there is (apparently) much more research directed towards vertex-coloring than edge-coloring? Prima facie, it seems that edge-coloring is just as "natural" a thing to investigate.
I can think of a few reasons:

Vertex coloring is well behaved under deletion and contraction of edges.
Vertex colorability is closely linked to the cycle matroid.
Edge-coloring can be regarded as vertex-coloring restricted to line graphs.
Since Vizing's theorem (that the chromatic index of $G$ is either $\Delta(G)$ or $\Delta(G)+1$) edge-coloring has been solved (asymptotically).

But is it really true that edge-coloring is less interesting than vertex-coloring?

REPLY [3 votes]: If you're going to be completely honest about the question, you need to consider edge colourings of multigraphs.  Even in this case there are not a lot of open problems.  Vizing's theorem already tells us that the chromatic index differs from the maximum degree by at most the maximum multiplicity.  However, more is known:

We can compute the fractional chromatic index in polynomial time; even approximating the fractional chromatic number of a graph is hard.
The fractional chromatic index and the chromatic index agree asymptotically, as first proven by Kahn.  However, the chromatic number is not even bounded by any function of the fractional chromatic number.
The Goldberg-Seymour conjecture implies that $\chi'_f$ and $\chi'$ differ by at most one.  This is a long-standing open problem on the chromatic index.  So what does that leave?  Well, proving that the difference is bounded by some universal constant, but not too much else.