• Source: 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.
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):





ch
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G
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2

χ
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G
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{\displaystyle \operatorname {ch} '(G)<2\chi '(G).}






ch
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K

n
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n


)
=
n
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{\displaystyle \operatorname {ch} '(K_{n,n})=n.}

This is the Dinitz conjecture, proven by Galvin (1995).





ch
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G
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1
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χ
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{\displaystyle \operatorname {ch} '(G)<(1+o(1))\chi '(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.




List coloring conjecture


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





ch
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G
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=

χ
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{\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 Kn,n.




References


Galvin, Fred (1995), "The list chromatic index of a bipartite multigraph", Journal of Combinatorial Theory, Series B, 63: 153–158, doi:10.1006/jctb.1995.1011.
Jensen, Tommy R.; Toft, Bjarne (1995), "12.20 List-Edge-Chromatic Numbers", Graph Coloring Problems, New York: Wiley-Interscience, pp. 201–202, ISBN 0-471-02865-7.
Kahn, Jeff (2000), "Asymptotics of the list chromatic index for multigraphs", Random Structures & Algorithms, 17 (2): 117–156, doi:10.1002/1098-2418(200009)17:2<117::AID-RSA3>3.0.CO;2-9

Kata Kunci Pencarian:

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