  We present a detailed quantum oscillatory study on the Dirac type-II
semimetallic candidates PdTe$_{2}$ and PtTe$_{2}$ \emph{via} the temperature
and the angular dependence of the de Haas-van Alphen (dHvA) and Shubnikov-de
Haas (SdH) effects. In high quality single crystals of both compounds, i.e.
displaying carrier mobilities between $10^3$ and $10^4$ cm$^2$/Vs, we observed
a large non-saturating magnetoresistivity (MR) which in PtTe$_2$ at a
temperature $T = 1.3$ K, leads to an increase in the resistivity up to $5
\times 10^{4}$ % under a magnetic field $\mu_0 H = 62$ T. These high mobilities
correlate with their light effective masses in the range of 0.04 to 1 bare
electron mass according to our measurements. For PdTe$_{2}$ the experimentally
determined Fermi surface cross-sectional areas show an excellent agreement with
those resulting from band-structure calculations. Surprisingly, this is not the
case for PtTe$_{2}$ whose agreement between calculations and experiments is
relatively poor even when electronic correlations are included in the
calculations. Therefore, our study provides a strong support for the existence
of a Dirac type-II node in PdTe$_2$ and probably also for PtTe$_2$. Band
structure calculations indicate that the topologically non-trivial bands of
PtTe$_2$ do not cross the Fermi-level ($\varepsilon_F$). In contrast, for
PdTe$_2$ the Dirac type-II cone does intersect $\varepsilon_F$, although our
calculations also indicate that the associated cyclotron orbit on the Fermi
surface is located in a distinct $k_z$ plane with respect to the one of the
Dirac type-II node. Therefore it should yield a trivial Berry-phase.
