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arXiv · 1302.2338

On-line list coloring of matroids

Michał Lasoń·Wojciech Lubawski
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Abstract

A coloring of a matroid is proper if elements of the same color form an independent set. A theorem of Seymour asserts that a k-colorable matroid is also colorable from any lists of size k. In this note we generalize this theorem to the on-line setting. We prove that a coloring of a matroid from lists of size k is possible even if appearances of colors in the lists are recovered color by color by an adversary, while our job is to assign a color immediately after it is recovered. We also prove a more general weighted version of our result with lists of varying sizes. In consequence we get a simple necessary and sufficient condition for matroid list colorability in general case. The main tool we use is the multiple basis exchange property, which we give a simple proof.

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Michał Lasoń, Wojciech Lubawski. 2013-02-10. On-line list coloring of matroids. https://doi.org/10.1016/j.dam.2016.08.002

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