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. Erd{o}s, 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 Ss,t on $s + t$ vertices whose centers have degrees s and t. Faudree, Faudree, Gould, and Jacobson determined that sat(n, St,t) = (t-1)n/2 + O(1). We show that sat(n,Ss,t) = (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.