The saturation number of wheels
Yanzhe Qiu, Zhen He, Mei Lu, Yiduo Xu
TL;DR
This work investigates the saturation and semi-saturation numbers of wheel graphs $W_k = K_1 \vee C_k$. By relating $W_k$-saturation to cycle-saturation via the presence of a conical vertex, the authors prove that for $k \ge 8$ and $n \ge 56k^3$, $sat(n,W_k) = ssat(n,W_k) = n-1 + sat(n-1,C_k)$, with all extremal graphs containing a conical vertex. They further develop a non-conical-vertex framework using a grand-connector structure to derive a general lower bound on $ssat^{\leq n-2}(n,W_k)$: $ssat^{\leq n-2}(n,W_k) \ge (\tfrac{5}{2} - \tfrac{3}{2k-2})n - \varphi(k)$ for $k \ge 6$, $n \ge 2k^3$, where $\varphi(k)$ is a specified rational function of $k$. The results reduce the problem to determining $ssat(n,C_k)$ for large $n$, and establish a robust methodology for tackling (semi)-saturation in join-graphs via structural decompositions and careful degree-based bounds. The paper closes with open questions about extending the exact equalities to smaller $k$ and clarifying the role of $ssat(n,C_k)$ in the full characterization of $sat(n,W_k)$ and $ssat(n,W_k)$.
Abstract
A graph $G$ is said to be $F$-free, if $G$ does not contain any copy of $F$. $G$ is said to be $F$-semi-saturated, if the addition of any nonedge $e \not \in E(G)$ would create a new copy of $F$ in $G+e$. $G$ is said to be $F$-saturated, if $G$ is $F$-free and $F$-semi-saturated. The saturation number $sat(n,F)$ (resp. semi-saturation number $ssat(n,F)$) is the minimum number of edges in an $F$-saturated (resp. $F$-semi-saturated) graph of order $n$. In this paper we proved several results on the (semi)-saturation number of the wheel graph $W_k=K_1 \vee C_k$. Let $k,n$ be positive integers with $k \geq 8$ and $n \geq 56k^3$, we showed that $(s)sat(n,W_k)=n-1+(s)sat(n-1,C_k)$. We also establish the lower bound of semi-saturation number of $W_k$ with restriction on maximum degree.