On the connectivity of certain graphs of high girth

Let q be a prime power and k ≥ 2 be an integer. In [2] and [3] it was determined that the number of components of certain graphs D(k, q) introduced in [1] is at least qt−1 where t = b k+2 4 c. This implied that these components (most often) provide the best-known asymptotic lower bound for the greatest number of edges in graphs of their order and girth. In [4], it was shown that the number of components is (exactly) qt−1 for q odd, but the method used there failed for q even. In this paper we prove that the number of components of D(k, q) for even q > 4 is again qt−1 where t = b k+2 4 c. Our proof is independent of the parity of q as long as q > 4. Furthermore, we show that for q = 4 and k ≥ 4, the number of components is qt.