Solving Graph Coloring Problems with Abstraction and Symmetry

TL;DR

This paper presents a novel methodology using abstraction and symmetry to solve complex graph coloring problems, providing new insights into the Ramsey number R(4,3,3) and narrowing down its possible configurations.

Contribution

The paper introduces a general approach based on abstraction and symmetry for solving hard graph edge-coloring problems, applied to the longstanding open problem of R(4,3,3).

Findings

Identified exactly 78,892 (3,3,3;13) Ramsey colorings.

Showed that a (4,3,3;30) Ramsey coloring, if it exists, must be (13,8,8) regular.

Conjectured that these results could lead to a proof that R(4,3,3)=30.

Abstract

This paper introduces a general methodology, based on abstraction and symmetry, that applies to solve hard graph edge-coloring problems and demonstrates its use to provide further evidence that the Ramsey number $R(4,3,3)=30$. 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. We illustrate our approach by showing that: (1) there are precisely 78{,}892 $(3,3,3;13)$ Ramsey colorings; and (2) if there exists a $(4,3,3;30)$ Ramsey coloring then it is (13,8,8) regular. Specifically each node has 13 edges in the first color, 8 in the second, and 8 in the third. We conjecture that these two results will help provide a proof that no $(4,3,3;30)$ Ramsey coloring exists implying that $R(4,3,3)=30$.