This book explores the problem of minimal valid graph coloring, first in the form of games and then of resolution algorithms. Emphasis is placed on deterministic, guaranteed and non-guaranteed methods. Stochastic methods are then just mentioned because already widely described in previous publications. The study then details a general quantum algorithm of polynomial complexity. A final chapter provides elements of reflection on diplomatic algorithms that, for the problem of coloring under resource constraints, seek a compromise minimizing frustrations. The appendix includes some mathematical additions and the source codes of the main algorithms presented, in particular the one of the quantum method.