A lower bound on the saturation number and a strengthening for triangle-free graphs

The saturation number sat (n, H) of a graph H and positive integer n is the minimum size of an n-vertex graph which does not contain a subgraph isomorphic to H but to which the addition of any edge creates such a subgraph. Erdos, Hajnal, and Moon first studied saturation numbers of complete graphs, and Cameron and Puleo introduced a general lower bound on sat (n, H). In this paper, we present another lower bound on sat (n, H), with a strengthening when H is triangle-free. Demonstrating its effectiveness, we determine the saturation numbers of diameter-3 trees up to an additive constant; these are double stars Sₒ, ₓ on s + t vertices whose centers have degrees s and t. Faudree, Faudree, Gould, and Jacobson determined that sat (n, Sₓ, ₓ) = (t-1) n/2 + O (1). We show that sat (n, Sₒ, ₓ) = (st+s) n/ (2t+4) + O (1) when s < t. We also determine lower and upper bounds on the saturation numbers of certain diameter-4 caterpillars.