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 they are 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. Maurice Clerc is a French mathematician. He worked in the Research and Development Department of France Télécom. Retired since 2004, he remains active in various fields of research, both in particle swarm optimization (PSO)-related fields and in other areas. He regularly publishes articles, gives keynote speeches in conferences, serves as a director and thesis jury, and occasionally works as an optimization consultant.