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Given a complete graph whose edges are colored by 3 colors, a compatible 3-coloring is a coloring of nodes such that no edge of the graph has the same color as its end-points.

The best algorithm I know is quasi-polynomial time. The problem is not known to be NP-complete. What is known about the status of this problem ?

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    $\begingroup$ Hi turkistany it would be very helpful if you were to provide more context for your questions, and this one in general. Specifically, what's a 'compatible' 3-coloring, and what's the history of the problem that you're already aware of ? $\endgroup$ – Suresh Venkat Aug 26 '10 at 19:30
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    $\begingroup$ The phrase "no edge of the graph has the same color as its end-points" might be somewhat ambiguous. What is meant is that for each edge, at least one of its endpoints has a different color than the edge. $\endgroup$ – Serge Gaspers Aug 26 '10 at 20:31
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Apparently, this paper by Cygan, Pilipczuk, Pilipczuk and Wojtaszczyk (2010+) gives a faster (polynomial time) algorithm: http://arxiv.org/abs/1004.5010

It was accepted to SODA 2011.

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The fastest known algorithm has running time $n^{O(\log n/ \log \log n)}$ [1].
Your question is reposed as an open problem in [2, page 4].

[1]: T. Feder, P. Hell, D. Král, and J. Sgall. Two algorithms for general list matrix partitions. SODA 2005, pages 870–876.

[2]: E. D. Demaine, M. Hajiaghayi, D. Marx. Open Problems -- Parameterized complexity and approximation algorithms. Dagstuhl Seminar 09511 (2009). Available here.

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