Deep neural networks for automatic image colorization often suffer from the color-bleeding artifact, a problematic color spreading near the boundaries between adjacent objects. Such color-bleeding artifacts debase the reality of generated outputs, limiting the applicability of colorization models in practice. Although previous approaches have attempted to address this problem in an automatic manner, they tend to work only in limited cases where a high contrast of gray-scale values are given in an input image. Alternatively, leveraging user interactions would be a promising approach for solving this color-breeding artifacts. In this paper, we propose a novel edge-enhancing network for the regions of interest via simple user scribbles indicating where to enhance. In addition, our method requires a minimal amount of effort from users for their satisfactory enhancement. Experimental results demonstrate that our interactive edge-enhancing approach effectively improves the color-bleeding artifacts compared to the existing baselines across various datasets.