Saturation numbers for disjoint stars

A graph G is called an H-saturated if G does not contain H as a subgraph, but the addition of any edge between two nonadjacent vertices in G results in a copy of H in G. The saturation number sat(n, H) is the minimum number of edges in G for all H-saturated graphs G of order n. For a graph F, let mF denote the disjoint union of m copies of F. In Faudree et al. (Electron J Combin 18:19-36, 2011) Faudree, Faudree and Schmitt proposed a problem that is to determine $$sat(n,mK_{1,k})$$ s a t ( n , m K 1 , k ) for all m and k. Let $$m\ge 2$$ m ≥ 2 , $$k\ge 4$$ k ≥ 4 and $$n\ge 3mk^2$$ n ≥ 3 m k 2 . In this paper, based on linear programming models, we show that $$\begin{aligned} \left\lceil \frac{n(k-1)-\lfloor \frac{k^2}{4}\rfloor }{2} \right\rceil +m-1 \le sat(n,mK_{1,k})\le \left\lceil \frac{n(k-1)-\lfloor \frac{k^2}{4}\rfloor +3m-1}{2} \right\rceil . \end{aligned}$$ n ( k - 1 ) - ⌊ k 2 4 ⌋ 2 + m - 1 ≤ s a t ( n , m K 1 , k ) ≤ n ( k - 1 ) - ⌊ k 2 4 ⌋ + 3 m - 1 2 .