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?