This paper deals with the difficult problem of video colorization. Methods in the literature are generally based on spatio-temporal video blocks, or on frame to frame color propagation methods, each technique having its own advantages and drawbacks. In this paper , we present both a novel automatic frame-to-frame propagation approach and an interactive correction method within a variational framework. The proposed method propagates colors from an initial colorized frame to the whole grayscale video sequence. The automatic propagation results may be visually unsuitable in some cases. To overcome this limitation, a spatio-temporal functional with a user-guided correction is introduced. Two primal-dual algorithms are designed to solve the proposed variational models. Numerical results show the efficiency and the potentiality of the proposed approach in comparison with state-of-the-art methods.