A computerised classification of some almost minimal triangle-free Ramsey graphs

Abstract

A graph $G$ is called a $(3,j;n)$-minimal Ramsey graph if it has the least amount of edges, $e(3,j;n)$, given that $G$ is triangle-free, the independence number $\alpha (G)<j$ and that $G$ has $n$ vertices. Triangle-free graphs $G$ with $\alpha (G)<j$ and where $e(G)-e(3,j;n)$ is small are said to be almost minimal Ramsey graphs. We look at a construction of some almost minimal Ramsey graphs, called $H_{13}$-patterned graphs. We make computer calculations of the number of almost minimal Ramsey triangle-free graphs that are $H_{13}$-patterned. The results of these calculations indicate that many of these graphs are in fact $H_{13}$-patterned. In particular, all but one of the connected $(3,j;n)$-minimal Ramsey graphs for $j\le 9$ are indeed $H_{13}$-patterned.