Breaking Symmetries in Graph Coloring Problems with Degree Matrices: the Ramsey Number R(4,3,3)=30?

Abstract. This paper introduces a general methodology that applies to solve graph edge-coloring problems and demonstrates its use to compute the Ramsey number R(4, 3, 3). The number R(4, 3, 3) is often presented as the unknown Ramsey number with the best chances of being found “soon”. Yet, its precise value has remained unknown for more than 50 years. The proposed technique is based on two well-studied concepts, abstraction and symmetry. First, we in-troduce an abstraction on graph colorings, degree matrices, that specify the de-gree of each vertex in each color. We compute, using a SAT solver, an over-approximation of the set of degree matrices of all solutions of the graph coloring problem. Then, for each degree matrix in the over-approximation, we compute, again using a SAT solver, the set of all solutions with matching degrees. Breaking symmetries, on degree matrices in the first step and with respect to graph isomor-phism in the second, is cardinal to the success of the approach. We illustrate the approach via two applications: proving that R(4, 3, 3) = 30 and computing the previously unknown number of (3, 3, 3; 13) Ramsey colorings (78,892).