Saturation numbers of $K_{2}\vee P_{k}$
Xiaoxue Zhang, Lihua You, Xinghui Zhao
TL;DR
This work determines the saturation numbers sat(n, K_{2}\vee P_k) for k\ge3, establishing the sharp formula sat(n, K_{2}\vee P_k)=2n-3+sat(n-2, P_k) for all sufficiently large n relative to k, via the parameter a_k. It also fully characterizes minimal K_{2}\vee P_k-saturated graphs: for n large, they are exactly K_{2}\vee F with F minimal P_k-saturated, while for small n they fall into a finite family \mathcal{G} of explicit constructions. The paper parallels and extends results on K_{1}\vee P_k and P_k-saturation, furnishing exact values for k=3,4,5 and several structural results that illuminate the role of vertex neighborhoods and diameter in saturated joins. The results have implications for the broader study of sat(n, K_s\vee P_k) and motivate further work on minimal P_k-saturated graphs for k\ge5 and larger joins. Overall, the authors deliver precise formulas, complete minimal-graph characterizations in key regimes, and a roadmap for future exploration of saturated joins.
Abstract
A graph $G$ is called $H$-saturated if $G$ contains no copy of $H$, but $G+e$ contains a copy of $H$ for any edge $e\in E(\overline{G})$. The saturation number of $H$ is the minimum number of edges in an $H$-saturated graph of order $n$, denoted by $sat(n,H)$. In this paper, we investigate $sat(n,K_{2}\vee P_{k})$, where $k\geq 3$. Let $a_k$ be an integer, defined as follows: $a_k=k$ for $3\leq k\leq 5$; $a_k=3\cdot 2^{t-1}-2$ for $k=2t\geq 6$; and $a_k=2^{t+1}-2$ for $k=2t+1\geq 7$. We show that $sat(n, K_{2}\vee P_{k})=2n-3+sat(n-2,P_{k})$ for $n\geq a_k+2$ and $k\geq 3$, characterize the $K_{2}\vee P_{k}$-saturated graphs with $sat(n,K_{2}\vee P_{k})$ edges, the $K_{1}\vee P_{k}$-saturated graphs with $sat(n,K_{1}\vee P_{k})$ edges for $3\leq k\leq5$ and the $P_{k}$-saturated graphs with $sat(n, P_{k})$ edges for $3\leq k\leq4$. Furthermore, we propose some questions for further research.