Some results on the saturation number of graphs

Abstract

For a given graph $F$, we say a (connected) graph $G$ is (connected) $F$-saturated if $G$ is $F$-free, and for any $e\in E\left(\overline{G}\right)$, $G+e$ creates an $F$-copy. The (connected) saturation number is the minimum number of edges of a (connected) $F$-saturated graph with $n$ vertices. We denote the saturation number and the connected saturation number by $sat(n,F)$ and $sa{t}_c(n,F)$, respectively. Evidently, $sat(n,F)\le sa{t}_c(n,F)$. A generalized friendship graph is regarded as the join of a clique and the union of some disjoint cliques. In this paper, we show the relationship of the saturation numbers between the unions of disjoint cliques and generalized friendship graphs, and as its application, obtain the saturation numbers of some generalized friendship graphs. And then we propose respectively two sufficient conditions for $sat(n,F)=sa{t}_c(n,F)$ and $sat(n,F)<sa{t}_c(n,F)$. In addition, we show $sa{t}_c(n,P_k\cup K_3)=n+2$ for sufficiently large $n$ and $k\ge 4$. Furthermore, we obtain an upper bound of the saturation number of $P_k\cup K_3$ for $k\ge 6$.