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

TL;DR

This paper investigates almost minimal triangle-free Ramsey graphs, focusing on a specific construction called $H_{13}$-patterned graphs, and provides computational evidence that most such graphs with certain parameters follow this pattern.

Contribution

It introduces the $H_{13}$-patterned construction for almost minimal Ramsey graphs and demonstrates through computer calculations that most graphs with certain parameters are $H_{13}$-patterned.

Findings

Most connected $(3,j;n)$-minimal Ramsey graphs for $j \,\leq\, 9$ are $H_{13}$-patterned.

Computer calculations show many almost minimal Ramsey graphs are $H_{13}$-patterned.

The study provides evidence for the prevalence of $H_{13}$-patterned structure in these 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 $j \leq 9$ are indeed $H_{13}$-patterned.