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Based on previously selected preliminary samples of Red Supergiants (RSGs) inM33 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 and420 RSGs in M31. With the time-series data from the iPTF survey spanning nearly2000 days, the period and amplitude of RSGs are analyzed. According to thelightcurves characteristics, they are classified into four categories in which84 and 56 objects in M33 and M31 respectively are semi-regular variables. Forthese semi-regular variables, the pulsation mode is identified by comparingwith the theoretical model, which yielded 19 (7) sources in the first overtonemode in M33 (M31), and the other 65 (49) RSGs in M33 (M31) in the fundamentalmode. The period-luminosity (P-L) relation is analyzed for the RSGs in thefundamental 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 inthe $V$ band, and the dust emission causes the less tight P-L relation in theSpitzer/[8.0] and [24] bands. The derived P-L relations in the 2MASS/$K_{\rmS}$ band are in agreement with those of RSGs in SMC, LMC and the Milky Waywithin the uncertainty range. It is found that the number ratio of RSGspulsating in the fundamental mode to the first overtone mode increases withmetallicity.

The Period-Luminosity Relations of Red Supergiants in M33 and M31