A star in spectral class K with a luminosity of \(10^6\) times that of the Sun would fall into the category of a red giant or possibly a red supergiant depending on other characteristics like size and temperature.
However, most red supergiants (like Betelgeuse) are considerably more luminous, often well over \(10^6\) solar luminosities. Red giants, on the other hand, usually have luminosities ranging from a few hundred to around \(10^5\) solar luminosities, but some can reach up to \(10^6\) solar luminosities as well. Given these points, a K class star with \(10^6\) luminosity is most consistent with being a red giant.
So, the best answer among the options provided is red giant.