  We revisit the K -band period-luminosity (P-L) relations of Galactic red
supergiants using Gaia Data Release 2 parallaxes and up to 70 yr of photometry
from AAVSO and ASAS campaigns. In addition, we examine 206 LMC red supergiants
using 50 yr of photometric data from the Digitised Harvard Astronomical Plate
Collection. We identified periods by computing power spectra and calculated the
period-luminosity relations of our samples and compared them with the
literature. Newly available data tighten the P-L relations substantially.
Identified periods form two groups: one with periods of 300-1000 days,
corresponding to pulsations, and another with Long Secondary Periods between
1000 and 8000 days. Among the 48 Galactic objects we find shorter periods in 25
stars and long secondary periods in 23 stars. In the LMC sample we identify 85
and 94 red supergiants with shorter and long secondary periods, respectively.
The P-L relation of the Galactic red supergiants is in agreement with the red
supergiants in both, the Large Magellanic Cloud and the Andromeda Galaxy. We
and no clear continuity between the known red giant period-luminosity
sequences, and the red supergiant sequences investigated here.
