Some results on minimum saturated graphs

TL;DR

This work investigates saturation numbers for graphs built from disjoint unions and joins, focusing on ${\rm sat}(n, {K_3, P_k})$ and the join ${K_1\lor F}$ with linear forests $F$. It develops a layered-tree framework and constructs two minimal ${\{K_3, P_k\}}$-saturated trees, $T_k^0$ and $T_k^1$, to prove that for $k\ge 10$ and large $n$, ${\rm sat}(n, {K_3, P_k}) = n - \left\lfloor \tfrac{n}{a_k^1} \right\rfloor$, where $a_k^1$ is given by the sizes of these trees. Using this, the paper derives upper and lower bounds for ${\rm sat}(n, K_3\cup P_k)$: $2 + {\rm sat}(n, {K_3, P_k}) \le {\rm sat}(n, K_3\cup P_k) \le 6 + {\rm sat}(n, {K_3, P_k})$, with the upper bound surpassing previous results. Additionally, the saturation number for the join ${K_1\lor F}$, with $F$ a linear forest without isolated vertices, is determined: ${\rm sat}(n, K_1\lor F) = (n-1) + {\rm sat}(n-1, F)$ for large $n$, and ${\rm Sat}(n, K_1\lor F) = \{K_1\lor H: H \text{ is a minimum } F\text{-saturated graph}\}$. The results advance understanding of how disjoint unions and joins influence saturation behavior and establish exact values under broad conditions, while also pointing to open questions for certain parameter regimes.

Abstract

Let $G$ be a graph and $\mathcal{F}$ be a family of graphs. We say a graph $G$ is $\mathcal{F}$-saturated if $G$ does not contain any member in $\mathcal{F}$ and for any $e\in E(\overline{G})$, $G+e$ creates a copy of some member in $ \mathcal{F}$. The saturation number of $\mathcal{F}$ is the minimum number of edges of an $\mathcal{F}$-saturated graphs with $n$ vertices, denoted by $\sat(n,\mathcal{F})$. If $\mathcal{F}=\{F\}$, then we write it as $\sat(n,F)$ for short. In this paper, we determine the exact value of $\sat(n,\{K_3,P_k\})$, and as its application, we obtain two bounds of $\sat(n,K_3\cup P_k)$ for $k\ge 10$ and sufficiently large $n$. Furthermore, $\sat(n,K_1\lor F)$ is determined, where $F$ is a linear forest without isolated vertices.

Figures (5)

• Figure 1: Two examples of $T_k$.
• Figure 2: Two examples of $T^0_k$.
• Figure 3: Two examples of $T^1_k$.
• Figure 4: An example of $Q_1$ for $k=9$.
• Figure 5: Graphs $T_1$, $T_2$, and $T_3$.

Theorems & Definitions (23)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Theorem 2.2
• Lemma 2.3
• Lemma 2.4
• Claim 1
• Claim 2
• Claim 3