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Recently it was shown that it is NP-hard to find a 5-colouring of a 3-colourable graph. More generally, it is NP-hard to distinguish $k$-colourable graphs from those that are not $(2k-1)$-colourable, for $k\ge 3$.

Turning the question around:

Is deciding if a 5-colourable graph is 3-colourable NP-hard?

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    $\begingroup$ It is not the decision problem that is hard; every 3-colorable graph is also 5-colorable. "Distinguish" is not the same as "decide". $\endgroup$ – Tom van der Zanden Nov 14 '19 at 16:56
  • $\begingroup$ Thanks, corrected. $\endgroup$ – András Salamon Nov 15 '19 at 6:39
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Yes, and this holds even for structured graphs. Indeed, every planar graph is 5-colorable (in fact even 4-colorable by the Four color theorem), but it is NP-complete to decide if a planar graph can be colored in 3 colors.

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