A star in spectral class K with a luminosity of \(10^6\) times that of the Sun is most likely a Red Giant.
- White Dwarf: These are small, dense stars that have lost their outer layers; they are not nearly as luminous.
- Red Giant: Stars in this category are large and cool, and they can have high luminosities, ranging from hundreds to millions of times that of the Sun.
- Red Supergiant: These stars are even larger than red giants but generally have spectral classes of M rather than K.
- Blue Giant: These stars are hot and luminous but are classified with spectral classes O or B, not K.
Given this information, the correct answer is B. Red Giant.