  About a year after core collapse supernova, dust starts to condense in the
ejecta. In meteorites, a fraction of C-rich presolar grains (e.g., silicon
carbide (SiC) grains of Type-X and low density graphites) are identified as
relics of these events, according to the anomalous isotopic abundances. Several
features of these abundances remain unexplained and challenge the understanding
of core-collapse supernovae explosions and nucleosynthesis. We show, for the
first time, that most of the measured C-rich grain abundances can be accounted
for in the C-rich material from explosive He burning in core-collapse
supernovae with high shock velocities and consequent high temperatures. The
inefficiency of the $^{12}$C($\alpha$,$\gamma$)$^{16}$O reaction relative to
the rest of the $\alpha$-capture chain at $T > 3.5\times10^8 \mathrm{K}$ causes
the deepest He-shell material to be carbon rich and silicon rich, and depleted
in oxygen. The isotopic ratio predictions in part of this material, defined
here as the C/Si zone, are in agreement with the grain data. The
high-temperature explosive conditions that our models reach at the bottom of
the He shell, can also be representative of the nucleosynthesis in hypernovae
or in the high-temperature tail of a distribution of conditions in asymmetric
supernovae. Finally, our predictions are consistent with the observation of
large $^{44}$Ca/$^{40}$Ca observed in the grains. This is due to the production
of $^{44}$Ti together with $^{40}$Ca in the C/Si zone, and/or to the strong
depletion of $^{40}$Ca by neutron captures.
