Minimum saturated graphs without $4$-cycles and $5$-cycles
Yue Ma
TL;DR
The paper determines the exact saturation number for the cycle family $\mathcal{C}_{\{4,5\}}$, showing $\mathrm{sat}(n,\mathcal{C}_{\{4,5\}})=\lceil\frac{5n}{4}-\frac{3}{2}\rceil$ for all $n$. It achieves this via a tight two-part approach: a constructive upper bound using a friendship-graph based $\mathcal{C}_{\{4,5\}}$-saturated graph on $n$ vertices with $|E|=\lceil\frac{5n}{4}-\frac{3}{2}\rceil$, and a matching lower bound derived from a minimal counterexample analyzed through degree-1 and degree-2 structures plus a discharging method. Core techniques include a BFS-tree framework, analysis of degenerated paths in the degree-2 regime, and a carefully designed discharging scheme that bounds the average degree. The work also surveys and extends understanding of saturation for cycle families, offering conjectures for broader $\mathcal{C}_I$-saturated graphs and presenting constructions like $J_{s,t}^{+r}$ to bound $\mathrm{sat}(n,\mathcal{C}_{a\mathbb{Z}_++2})$ exactly.
Abstract
Given a family of graphs $\mathcal{F}$, a graph $G$ is said to be $\mathcal{F}$-saturated if $G$ does not contain a copy of $F$ as a subgraph for any $F\in\mathcal{F}$, but the addition of any edge $e\notin E(G)$ creates at least one copy of some $F\in\mathcal{F}$ within $G$. The minimum size of an $\mathcal{F}$-saturated graph on $n$ vertices is called the saturation number, denoted by $\mbox{sat}(n, \mathcal{F})$. Let $C_r$ be the cycle of length $r$. In this paper, we study on $\mbox{sat}(n, \mathcal{F})$ when $\mathcal{F}$ is a family of cycles. In particular, we determine that $\mbox{sat}(n, \{C_4,C_5\})=\lceil\frac{5n}{4}-\frac{3}{2}\rceil$ for any positive integer $n$.