Some results on the saturation number for unions of cliques

Abstract

Graph $G$ is $H$-saturated if $H$ is not a subgraph of $G$ and $H$ is a subgraph of $G+e$ for any edge $e$ not in $G$. The saturation number for a graph $H$ is the minimal number of edges in any $H$-saturated graph of order $n$. In this paper, the saturation number for $K_p\cup (t-1)K_q$ ($t\geqslant 3$ and $2\leqslant p<q$) is determined, and the extremal graph for $K_p\cup 2K_q$ is determined. Moreover, the saturation number and the extremal graph for $K_p\cup K_q\cup K_r$ ($ r\geqslant p+q$) are completely determined.