  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).
