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This paper proposes a novel methodology to solve the graph coloring problem (GCP) using the Q'tron neural- network (NN) model. The Q'tron NN for GCP will be built as a known-energy system. This can make the NN local- minima-free and perform the so-called goal-directed search. Consider k-GCP as a goal to solve a GCP using at most k different colors. By continuously refining our goal, i.e., decreasing the value k, we can then 'demand' the NN to fulfill better and better goals progressively. Experiments using DI-MACS benchmarks were done using such an approach, and comparison was made with the DSATUR algorithm. The result supports the soundness of our approach.