This paper studies neural algorithmic reasoning for approximate k-coloring, a relaxed graph-coloring problem that seeks to minimize conflicts while using at most k colors. It improves a differentiable graph neural network approach through orthogonal node-feature initialization and a degree-aware conflict loss, then introduces a lightweight greedy local search method. A recursive warm-start strategy, based on solving a (k-1)-coloring before the k-coloring task, improves both the local search and GNN approaches. Experiments across graph families show that local search is strongest on small graphs, while the GNN scales better to larger inputs.