The saturation number of W 4

TL;DR

This work resolves the exact saturation number for wheel graphs $W_4$ by proving ${\rm sat}(n,W_4)=\left\lfloor\frac{5n-10}{2}\right\rfloor$ for all $n\ge 6$ and by fully characterizing the extremal $W_4$-saturated graphs. The authors develop a structural framework for $W_4$-saturated graphs, introducing a minimum-degree vertex analysis, the sets $V_i$ with respect to $N(x)$, and the shadow/charging tools, then apply a comprehensive case analysis (based on $\delta(G)$ and $e(N[x])$) together with a discharging method to bound the edge count. The main result is achieved by isolating the extremal configurations into explicit graph families, namely $\mathcal{A}_n^1,\mathcal{A}_n^2,\mathcal{A}_n^3$ and $\mathcal{B}_n^1,\mathcal{B}_n^2,\mathcal{B}_n^3$, with a parity-dependent description of Sat$(n,W_4)$. The findings extend the classic saturation theory for wheels beyond $W_3$, and provide a complete classification of extremal graphs in this case, leveraging both constructive upper bounds and intricate lower-bound discharging arguments.

Abstract

For a fixed graph $H$, a graph $G$ is called $H$-saturated if $G$ does not contain $H$ as a (not necessarily induced) subgraph, but $G+e$ contains a copy of $H$ for any $e\in E(\overline{G})$. The saturation number of $H$, denoted by ${\rm sat}(n,H)$, is the minimum number of edges in an $n$-vertex $H$-saturated graph. A wheel $W_n$ is a graph obtained from a cycle of length $n$ by adding a new vertex and joining it to every vertex of the cycle. A well-known result of Erdős, Hajnal and Moon shows that ${\rm sat}(n,W_3)=2n-3$ for all $n\geq 4$ and $K_2\vee \overline{K_{n-2}}$ is the unique extremal graph, where $\vee$ denotes the graph join operation. In this paper, we study the saturation number of $W_4$. We prove that ${\rm sat}(n,W_4)=\lfloor\frac{5n-10}{2}\rfloor$ for all $n\geq 6$ and give a complete characterization of the extremal graphs.

Figures (9)

• Figure 1: The four graphs with $5$ vertices and $7$ edges.
• Figure 2: The extremal graph $H^*$.
• Figure 3: The graph families $\mathcal{F}_n^1$, $\mathcal{F}_n^2$ and $\mathcal{F}_n^3$.
• Figure 4: The graph families $\mathcal{B}_n^1$, $\mathcal{B}_n^2$ and $\mathcal{B}_n^3$.
• Figure 5: The configuration in Lemma \ref{['lem3.2']}.

Theorems & Definitions (38)

• Theorem 1.1
• Theorem 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• Proposition 2.4
• Proposition 2.5
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3