Saturation numbers for disjoint stars

Abstract

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 $sat(n,mK_{1,k})$ for all m and k. Let $m\ge 2$, $k\ge 4$ and $n\ge 3mk^2$. In this paper, based on linear programming models, we show that

$$\begin{aligned} \left\lceil \frac{n(k-1)-\lfloor \frac{k^2}{4}\rfloor }{2} \right\rceil +m-1 \le sat(n,mK_{1,k})\le \left\lceil \frac{n(k-1)-\lfloor \frac{k^2}{4}\rfloor +3m-1}{2} \right\rceil . \end{aligned}$$