The VMC Survey -- XXXVII. Pulsation periods of dust enshrouded AGB stars in the Magellanic Clouds
(abridged)Variability is a key property of stars on the asymptotic giant branch (AGB). Their pulsation period is related to the luminosity and mass-loss rate of the star. The long-period variables (LPVs) and Mira variables are the most prominent of all types of variability of evolved stars. The reddest, most obscured AGB stars are too faint in the optical and have eluded large variability surveys. Selection criteria are derived based on colour-colour and colour-magnitude diagrams from the combination of VISTA Magellanic Cloud (VMC) survey, Spitzer IRAC and AllWISE data. After eliminating LPVs with known periods shorter than 450 days, a sample of 1299 candidate obscured AGB stars is selected. K-band LCs are constructed combining the epoch photometry available in the VMC survey with literature data, analysed for variability and fitted with a single period sine curve to derive mean magnitudes, amplitudes and periods. A subset of 254 stars are either new variables, or known variables where the period we find is better determined than the literature value, or longer than 1000 days. The spectral energy distributions (SEDs) of these stars are fitted to a large number of templates. For this purpose the SEDs and Spitzer IRS spectra of some non-AGB stars are also fitted to have templates of the most likely contaminants in the sample. A sample of 217 likely LPVs is found. Thirty-four stars have periods longer than 1000 days although some of them have alternative shorter periods. Mass-loss rates and luminosities are estimated from the template fitting. Period-luminosity relations are presented for C- and O-rich Miras that appear to be extensions of relations derived in the literature for shorter periods. The fit for the C-stars is particularly well defined (with 182 objects) and reads Mbol = (-2.27 \pm 0.20) log P + (1.45 \pm 0.54)mag with an rms of 0.41 mag.