This article demonstrates how physics-inspired graph neural networks solve the graph coloring problem, a fundamental NP-hard optimization challenge with real-world applications.

• Graph coloring assigns labels to graph vertices ensuring adjacent vertices have different colors
• Problem relates to Potts model from statistical physics, providing natural loss function
• Physics-inspired GNNs frame coloring as multi-class node classification without penalty terms
• Approach scales to graphs with ~20,000 nodes, outperforming traditional algorithms on dense instances
• Achieves sub-one-percent normalized errors across benchmark COLOR instances
• Scheduling use case demonstrates practical application: assigning resources to overlapping time requests
• Offers alternative to quantum computing approaches requiring q×n binary variables
• Framework generalizes to community detection, clustering, and minimum clique cover problems

Physics-inspired GNNs provide scalable, practical solutions to graph coloring while building quantum computing literacy for enterprise customers.