  Period-luminosity (PL) relations of variable red giants in the Large (LMC)
and Small Magellanic Clouds (SMC) are presented. The PL diagrams are plotted in
three planes: logP-K_S, logP-W_{JK}, and logP-W_I. Fourteen PL sequences are
distinguishable, and some of them consist of three closely spaced ridges. Each
of the sequences is fitted with a linear or quadratic function. The
similarities and differences between the PL relations in both galaxies are
discussed for four types of red giant variability: OGLE Small Amplitude Red
Giants (OSARGs), Miras and Semiregular Variables (SRVs), Long Secondary Periods
(LSPs) and ellipsoidal variables. We propose a new method of separating OSARGs
from non-variable stars and SRVs. The method employs the position in the
reddening-free PL diagrams and the characteristic period ratios of these
multiperiodic variables. The PL relations for the LMC OSARG are compared with
the calculated relations for RGB models along isochrones of relevant ages and
metallicities. We also compare measured periods and amplitudes of the OSARGs
with predictions based on the relations valid for less luminous solar-like
pulsators. Miras and SRVs seem to follow PL relation of the same slopes in the
LMC and SMC, while for LSP and ellipsoidal variables slopes in both galaxies
are different. The PL sequences defined by LSP variables and binary systems
overlap in the whole range of analyzed wavebands. We put forward new arguments
for the binary star scenario as an explanation of the LSP variability and
elaborate on it further. The measured pulsation to orbital period ratio implies
nearly constant ratio of the star radius to orbital distance, R/A=0.4, as we
find. Combined effect of tidal friction and mass loss enhanced by the low-mass
companion may explain why such a value is preferred.
