  Pulsating white dwarfs provide constraints to the evolution of progenitor
stars. We revise He-burning stellar models, with particular attention to core
convection and to its connection with the nuclear reactions powering energy
generation and chemical evolution Theoretical results are compared to the
available measurements for the variable white dwarf GD 358, which indicate a
rather large abundance of central oxygen. We show that the attempt to constrain
the relevant nuclear reaction rate by means of the white dwarf composition is
faced with a large degree of uncertainty related to evaluating the efficiency
of convection-induced mixing.By combining the uncertainty of the convection
theory with the error on the relevant reaction rate we derive that the present
theoretical prediction for the central oxygen mass fraction in white dwarfs
varies between 0.3 and 0.9. Unlike previous claims, we find that models taking
into account semiconvection and a moderate C12(alpha,gamma)O16 reaction rate
are able to account for a high central oxygen abundance. The rate of the
C12(alpha,gamma)O16 used in these models agrees with the one recently obtained
in laboratory experiments (Kunz et al. 2002). On the other hand, when
semiconvection is inhibited, as in the case of classical models (bare
Schwarzschild criterion) or in models with mechanical overshoot, an extremely
high rate of the C12(\alpha,\gamma)O16 reaction is needed to account for a
large oxygen production. Finally, we show that the apparent discrepancy between
our result and those reported in previous studies depends on the method used to
avoid the convective runaways (the so called breathing pulses), which are
usually encountered in modeling late stage of core He-burning phase.
