Tight Upper Bounds for the Domination Numbers of Graphs with Given Order and Minimum Degree, II

Let γ(n, δ) denote the largest possible domination number for a graph of order n and minimum degree δ. This paper is concerned with the behavior of the right side of the sequence n = γ(n, 0) ≥ γ(n, 1) ≥ · · · ≥ γ(n, n− 1) = 1. We set δk(n) = max{δ | γ(n, δ) ≥ k}, k ≥ 1. Our main result is that for any fixed k ≥ 2 there is a constant ck such that for sufficiently large n, n− ckn ≤ δk+1(n) ≤ n− n(k−1)/k. The lower bound is obtained by use of circulant graphs. We also show that for n sufficiently large relative to k, γ(n, δk(n)) = k. The case k = 3 is examined in further detail. The existence of circulant graphs with domination number greater than 2 is related to a kind of difference set in Zn. 2000 Mathematics Subject Classifications: Primary 05C69, Secondary 05C35