List edge-coloring

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"):
# ch&prime;("G") < 2 χ&prime;("G").
# ch&prime;(K"n","n") = "n". (Galvin 1995)
# ch&prime;("G") < (1 + o(1))χ&prime;("G"), i.e. the list chromatic index and the chromatic index agree asymptotically. (Kahn 2000)Here χ&prime;("G") is the chromatic index of "G"; and K"n","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&prime;("G") = χ&prime;("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.

See also

* List coloring
* Edge coloring

References

* Galvin, Fred (1995). The list chromatic index of a bipartite multigraph. "J. Combin. Theory (B)" 63, 153&ndash;158.
* Jensen, Tommy R.; Toft, Bjarne (1995). "Graph coloring problems". New York: Wiley-Interscience. ISBN 0-471-02865-7.
* Kahn, Jeff (2000). Asymptotics of the list chromatic index for multigraphs. "Rand. Struct. Alg." 17, 117&ndash;156.


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