Definitions

# List edge-coloring

In mathematics, list edge-coloring is a type of graph coloring. More precisely, a list edge-coloring is a choice function that maps every edge to a color from a prescribed list for that edge. A graph is k-edge-choosable if it has a proper list edge-coloring - one in which no two adjacent edges receive the same color - for every collection of lists of k colors. The edge choosability, or list edge colorability, list edge chromatic number, or list chromatic index, ch′(G) of a graph G is the least number k such that G is k-edge-choosable.

Some properties of ch′(G):

1. ch(G) < 2 χ(G).
2. ch(Kn,n) = n. (Galvin 1995)
3. ch(G) < (1 + o(1))χ(G), i.e. the list chromatic index and the chromatic index agree asymptotically. (Kahn 2000)

Here χ(G) is the chromatic index of G; and Kn,n, the complete bipartite graph with equal partite sets.

The most famous open problem about list edge-coloring is probably the list coloring conjecture.

List coloring conjecture.

ch(G) = χ(G).

This conjecture has a fuzzy origin. Interested readers are directed to [Jensen, Toft 1995] for an overview of its history. It is also a generalization of the longstanding Dinitz conjecture, which was eventually solved by Galvin in 1995 using list edge-coloring.