A computerised classification of some almost minimal triangle-free Ramsey graphs
Oliver Kr\"uger

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 is called a -minimal Ramsey graph if it has the least amount of edges, , given that is triangle-free, the independence number and that has vertices. Triangle-free graphs with and where is small are said to be almost minimal Ramsey graphs. We look at a construction of some almost minimal Ramsey graphs, called -patterned graphs. We make computer calculations of the number of almost minimal Ramsey triangle-free graphs that are -patterned. The results of these calculations indicate that many of these graphs are in fact -patterned. In particular, all but one of the connected -minimal Ramsey graphs for are indeed -patterned.
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Taxonomy
TopicsLimits and Structures in Graph Theory · Advanced Graph Theory Research · Complexity and Algorithms in Graphs
