报告题目：Hamilton cycles in $1$-tough and $(P_2\cup kP_1)$-free graphs

报告人：胡智全 教授

报告时间：2026年1月10日9:00-10:00

报告地点：科技园阳光楼南815

邀请人：林启忠

邀请单位：福州大学数学与统计学院

报告内容简介：For a graph $G$, define $\mu_k(G):=\min~\{\max_{x\in S}d_G(x):~S\in \mathcal{S}_k\}$, where $\mathcal{S}_k$ is the set consisting of all independent sets $\{u_1,\ldots,u_k\}$ of $G$ such that some vertex, say $u_i$ ($1\leq i\leq k$), is at distance two from every other vertex in it. A graph $G$ is called $1$-tough if for each cut set $S\subseteq V(G)$, $G-S$ has no more than $|S|$ components. Recently, Shi and Shan \cite{Shi} conjectured that for each integer $k\geq 4$, being $2k$-connected is sufficient for $1$-tough $(P_2\cup kP_1)$-free graphs to be hamiltonian, which was confirmed by Xu et al. [Discrete Math. 347 (2024) 113755] and Ota and Sanka [Discrete Math. 347 (2024) 113841], respectively. In this talk, we show the following Fan-type theorem: Let $k\geq 2$ be an integer and let $G$ be a $1$-tough and $k$-connected $(P_2\cup kP_1)$-free graph with $\mu_{k+1}(G)\geq\frac{7k-6}{5}$, then $G$ is hamiltonian or the Petersen graph.

报告人简介：胡智全，华中师范大学教授、博士生导师，主要从事结构图论研究。2001年中国科学院数学与系统科学研究院博士研究生毕业，2002至2003年在巴黎十一大计算机实验室从事博士后研究。多次访问香港大学和美国佐治亚州立大学。现任中国组合数学与图论学会理事。主持国家自然科学基金面上项目5项，在Journal of Combinatorial Theorey Series B, Journal of Graph Theory, SIAM J. on Discrete Math等杂志发表论文40余篇。