A Dynamic Extension of the Petford–Welsh Coloring Algorithm for Estimating the Chromatic Number

We introduce a dynamic extension of the Petford–Welsh coloring algorithm that estimates the chromatic number of a graph without requiring k as an input. The algorithm is based on a model closely related to Boltzmann machines that minimize the Ising Hamiltonian. Starting from a minimal coloring, it adaptively adjusts the number of colors according to solution quality.

We evaluate the method on graphs from the DIMACS benchmark suite under various initialization strategies. The results indicate that the algorithm achieves near-optimal solutions with high robustness and shows promising effectiveness on real-world instances. While additional adaptive or problem-specific strategies may be necessary for harder cases, the main advantage of the approach lies in its inherent parallelism, which opens avenues for further exploration.