In this work, we present a complete hybrid classical-quantum algorithm involving a quantum sampler based on neutral atom platforms. This approach is inspired by classical column generation frameworks developed in the field of Operations Research and shows how quantum procedures can assist classical solvers in addressing hard combinatorial problems. We benchmark our method on the Minimum Vertex Coloring problem and show that the proposed hybrid quantum-classical column generation algorithm can yield good solutions in relatively few iterations. We compare our results with state-of-the-art classical and quantum approaches.