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

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... ns 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.