Andrásfai graph

In graph theory, an Andrásfai graph is a triangle-free circulant graph named after Béla Andrásfai.
Properties

The Andrásfai graph And(n) for any natural number \( n \geq 1 \) is a circulant graph on \( {\displaystyle 3n-1} \) vertices, in which vertex k is connected by an edge to vertices \( {\displaystyle k\pm j,} \) for every j that is congruent to 1 mod 3. For instance, the Wagner graph is an Andrásfai graph, the graph And(3).

The graph family is triangle-free, and And(n) has an independence number of n. From this the formula \( {\displaystyle R(3,n)\geq 3(n-1)} \) results, where R(n,k) is the Ramsey number. The equality holds for \( {\displaystyle n=3,n=4} \) only.

References

Godsil, Chris; Royle, Gordon F. (2013) [2001]. "§6.10–6.12: The Andrásfai Graphs—Andrásfai Coloring Graphs, A Characterization". Algebraic Graph Theory. Graduate Texts in Mathematics. 207. Springer. pp. 118–123. ISBN 978-1-4613-0163-9.
Andrásfai, Béla (1971). Ismerkedés a gráfelmélettel (in Hungarian). Budapest: Tankönyvkiadó. pp. 132–5. OCLC 908973331.
Weisstein, Eric W. "Andrásfai Graph". MathWorld.