each vertex in the coloring must have 13 edges in the first color, and 8 edges in each of the other two colors. This result is already considered significant progress in the research on Ramsey numbers as stated in [@XuRad2015]. To further apply this technique to determine if there exists a ${\langle 13,8,8 \rangle}$ regular $(4,3,3;30)$ Ramsey coloring requires to first compute the currently unknown set ${{\cal R}}(3,3,3;13)$.

Sections \[sec:symBreak\]—\[sec:33313b\] present the second step: computing ${{\cal R}}(3,3,3;13)$. Section \[sec:symBreak\] illustrates how a straightforward approach, combining SAT solving with *symmetry breaking*, works for smaller instances but not for ${{\cal R}}(3,3,3;13)$. Then Section \[sec:abs\] introduces an *abstraction*, called degree matrices, Section \[sec:33313\] demonstrates how to compute degree matrices for ${{\cal R}}(3,3,3;13)$, and Section \[sec:33313b\] shows how to use the degree matrices to compute ${{\cal R}}(3,3,3;13)$. Section \[sec:433\_30\] presents the third step re-examining the embedding technique described in Section \[sec:embed\] which given the set ${{\cal R}}(3,3,3;13)$ applies to prove that there does not exist any $(4,3,3;30)$ Ramsey coloring which is also ${\langle 13,8,8 \rangle}$ regular. Section \[sec:conclude\] presents a conclusion.

Preliminaries and Notation {#sec:prelim}
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In this paper, graphs are always simple, i.e. undirected and with no self loops. For a natural number 