Universal Vertices and Saturation Numbers for Disjoint Triangles

Abstract

A graph $G$ is $\mathcal F$-saturated if $G$ contains no member of $\mathcal F$, but the addition of any non-edge creates a copy of a member of $\mathcal F$. For $m\ge 1$, let $(m+1)K_3$ denote the vertex-disjoint union of $(m+1)$ triangles. In this paper, we study $(m+1)K_3$-saturated graphs. We construct a family of $(m+1)K_3$-saturated graphs which gives the uniform upper bound $\operatorname{sat}((m+1)K_3,n)=O(n^{3/2})$ for all $m\ge 1$ and $n\ge 3m+3$. For general $(m+1)K_p$-saturated graphs, Faudree et al. determined $\operatorname{sat}((m+1)K_p, n)$ for sufficiently large $n$. In the case of $p=3$, we lower Faudree's threshold from $n \ge 12m + 3$ to $n\ge 9m+5$ and prove that $\operatorname{sat}((m+1)K_3,n)=n+6m-1.$ We also provide two structural restrictions on the components obtained after deleting all vertices of degree $|V(G)|-1$ from an $(m+1)K_3$-saturated graph.