List edge-coloring


In mathematics, list edge-coloring is a type of graph coloring that combines list coloring and edge coloring.
An instance of a list edge-coloring problem consists of a graph together with a list of allowed colors for each edge. A list edge-coloring is a choice of a color for each edge, from its list of allowed colors; a coloring is proper if no two adjacent edges receive the same color.
A graph G is k-edge-choosable if every instance of list edge-coloring that has G as its underlying graph and that provides at least k allowed colors for each edge of G has a proper coloring.
The edge choosability, or list edge colorability, list edge chromatic number, or list chromatic index, ch′ of graph G is the least number k such that G is k-edge-choosable. It is conjectured that it always equals the chromatic index.

Properties

Some properties of ch′:
  1. ch′ < 2 χ′.
  2. ch′ = n. This is the Dinitz conjecture, proven by.
  3. ch′ < χ′, i.e. the list chromatic index and the chromatic index agree asymptotically.
Here χ′ is the chromatic index of G; and Kn,n, the complete bipartite graph with equal partite sets.

List coloring conjecture

The most famous open problem about list edge-coloring is probably the list coloring conjecture.
This conjecture has a fuzzy origin; overview its history. The Dinitz conjecture, proven by, is the special case of the list coloring conjecture for the complete bipartite graphs Kn,n.