A star in spectral class K with a luminosity of \(10^6\) times that of the Sun (or \(10^6 L_{\odot}\)) would typically be classified as a red giant or red supergiant.
However, given the options you provided, the most appropriate choice is:
c. red giant
While red supergiants can also possess high luminosity, they are typically classified as spectral types earlier than K (usually M). Therefore, a K-class star with a luminosity of \(10^6 L_{\odot}\) fits best with the characteristics of a red giant.