  Based on previously selected preliminary samples of Red Supergiants (RSGs) in
M33 and M31, the foreground stars and luminous Asymptotic Giant Branch stars
(AGBs) are further excluded, which leads to the samples of 717 RSGs in M33 and
420 RSGs in M31. With the time-series data from the iPTF survey spanning nearly
2000 days, the period and amplitude of RSGs are analyzed. According to the
lightcurves characteristics, they are classified into four categories in which
84 and 56 objects in M33 and M31 respectively are semi-regular variables. For
these semi-regular variables, the pulsation mode is identified by comparing
with the theoretical model, which yielded 19 (7) sources in the first overtone
mode in M33 (M31), and the other 65 (49) RSGs in M33 (M31) in the fundamental
mode. The period-luminosity (P-L) relation is analyzed for the RSGs in the
fundamental mode. It is found the P-L relation is tight in the infrared, i.e.
the 2MASS $JHK_{\rm S}$ bands and the short-wavelength bands of Spitzer.
Meanwhile, the inhomogeneous extinction causes the P-L relation scattering in
the $V$ band, and the dust emission causes the less tight P-L relation in the
Spitzer/[8.0] and [24] bands. The derived P-L relations in the 2MASS/$K_{\rm
S}$ band are in agreement with those of RSGs in SMC, LMC and the Milky Way
within the uncertainty range. It is found that the number ratio of RSGs
pulsating in the fundamental mode to the first overtone mode increases with
metallicity.
