Solving Graph Coloring Problems with Abstraction and Symmetry: the Ramsey Number R(4, 3, 3)=30
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.