List edge-coloring

In graph theory, 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. In other words, when the list for each edge has length k, no matter which colors are put in each list, a color can be selected from each list so that G is properly colored. The edge choosability, or list edge colorability, list edge chromatic number, or list chromatic index, ch'(G) 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.

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

Some properties of ch'(G):

1. ${\displaystyle \operatorname {ch} '(G)<2\chi '(G).}$
2. ${\displaystyle \operatorname {ch} '(K_{n,n})=n.}$ This is the Dinitz conjecture, proven by Galvin (1995).
3. ${\displaystyle \operatorname {ch} '(G)<(1+o(1))\chi '(G),}$ i.e. the list chromatic index and the chromatic index agree asymptotically (Kahn 2000).
4. For bipartite graphs, ${\displaystyle ch'(G)=\chi '(G)}$.¹ For every simple bipartite graph, ${\displaystyle ch'(G)=\Delta }$.²

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

$${\displaystyle \operatorname {ch} '(G)=\chi '(G).}$$

This conjecture has a fuzzy origin; Jensen & Toft (1995) overview its history. The Dinitz conjecture, proven by Galvin (1995), is the special case of the list coloring conjecture for the complete bipartite graphs K_n,n.