Domination, radius, and minimum degree

We prove sharp bounds concerning domination number, radius, order and minimum degree of a graph. In particular, we prove that if G is a connected graph of order n, domination number γ and radius r, then frac(2, 3) r ≤ γ ≤ n - frac(4, 3) r + frac(2, 3). Equality is achieved in the upper bound if, and only if, G is a path or a cycle on n vertices with n ≡ 4 (mod 6). Further, if G has minimum degree δ ≥ 3 and r ≥ 6, then using a result due to Erdös, Pach, Pollack, and Tuza [P. Erdös, J. Pach, R. Pollack, Z. Tuza, Radius, diameter, and minimum degree. J. Combin. Theory B 47 (1989), 73-79] we show that γ ≤ n - frac(2, 3) (r - 6) δ.