Explicit tough Ramsey graphs

TL;DR

This paper explores the properties of certain finite field graphs, demonstrating they serve as counterexamples to Chvatal's conjecture on t-tough graphs and offer bounds for the Ramsey number R(3,k).

Contribution

It introduces new counterexamples to Chvatal's conjecture using finite field graphs and establishes a constructive lower bound for the Ramsey number R(3,k).

Findings

Finite field graphs are counterexamples to Chvatal's conjecture.

These graphs are Ramanujan or asymptotically Ramanujan for large q.

They provide a lower bound for the Ramsey number R(3,k).

Abstract

A graph G is t-tough if any induced subgraph of it with x > 1 connected components is obtained from G by deleting at least tx vertices. Chvatal conjectured that there exists an absolute constant t_0 so that every t_0-tough graph is pancyclic. This conjecture was disproved by Bauer, van den Heuvel and Schmeichel by constructing a t_0-tough triangle-free graph for every real t_0. For each finite field F_q with q odd, we consider graphs associated to the finite Euclidean plane and the finite upper half plane over F_q. These graphs have received serious attention as they have been shown to be Ramanujan (or asymptotically Ramanujan) for large q. We will show that for infinitely many q, these graphs provide further counterexamples to Chvatal's conjecture. They also provide a good constructive lower bound for the Ramsey number R(3,k).