List edge-coloring explained

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(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.

Properties

Some properties of ch(G):

1. ch(G) < 2 χ(G).
2. ch(K_n,n) = n. This is the Dinitz conjecture, proven by .
3. ch(G) < (1 + o(1))χ(G), i.e. the list chromatic index and the chromatic index agree asymptotically .

Here χ(G) is the chromatic index of G; and K_n,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.

ch(G) = χ(G).

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 K_n,n.

References

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