Sketch colorization is usually expensive and time-consuming for artists, and automating this process can have many pragmatic applications in the animation, comic book, and video game industry. However, automatic image colorization faces many challenges, because sketches not only lack texture information but also potentially entail complicated objects that require acute coloring. These difficulties usually result in incorrect color assignments that can ruin the aesthetic appeal of the final output. In this paper, we present a novel component-based matching framework that combines deep learned features and quadratic programming {\color{red} with a new cost function} to solve this colorization problem. The proposed framework inputs a character's sketches as well as a colored image in the same cut of a movie, and outputs a high-quality sequence of colorized frames based on the color assignment in the reference colored image. To carry out this colorization task, we first utilize a pretrained ResNet-34 model to extract elementary components' features to match certain pairs of components (one component from the sketch and one from reference). Next, a graph representation is constructed in order to process and match the remaining components that could not be done in the first step. Since the first step has reduced the number of components to be matched by the graph, we can solve this graph problem in a short computing time even when there are hundreds of different components present in each sketch. We demonstrate the effectiveness of the proposed solution by conducting comprehensive experiments and producing aesthetically pleasing results. To the best of our knowledge, our framework is the first work that combines deep learning and graph representation to colorize anime and achieves a high pixel-level accuracy at a reasonable time cost.