On the saturation spectrum of the unions of disjoint cycles

Abstract

Let G be a graph and $H$ be a family of graphs. We say G is $H$-saturated if G does not contain a copy of H with $H\in H$, but the addition of any edge $e\notin E\left(G\right)$ creates at least one copy of some $H\in H$ within $G+e$. The saturation number of $H$ is the minimum size of an $H$-saturated graph on n vertices, and the saturation spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on n vertices. Let $k{C}_{\ge 3}$ be the family of the unions of k vertex-disjoint cycles. In this note, we completely determine the saturation number and the saturation spectrum of $k{C}_{\ge 3}$ for $k=2$ and give some results for $k\ge 3$.