List edge-coloring
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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 list edge-coloring 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):
- ch′(G) < 2 χ′(G).
- ch′(Kn,n) = n. (Galvin 1995)
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.
References
- Galvin, Fred (1995). The list chromatic index of a bipartite multigraph. J. Combin. Theory (B) 63, 153–158.
- Jensen, Tommy R.; Toft, Bjarne (1995). Graph coloring problems. New York: Wiley-Interscience. ISBN 0-471-02865-7.