勷勤数学•专家报告

题      目：Hamilton-Connected Hourglass-free Line Graphs

报  告  人： 赖虹建 教授  (邀请人：尤利华)

                                     西弗吉尼亚大学

时      间： 1月30日  10：30-11:30

地     点：数科院东楼401

报告人简介:

        美国西弗吉尼亚大学数学系终身教授，博士生导师，国际知名的图论专家。主要研究领域包括图论中的欧拉子图、哈密尔顿性问题、整数流以及图论中的染色和连通度问题，出版学术著作两部，发表学术论文300余篇。完成了两部专著：由克鲁亚学术出版社（Kluwer Academic Publishing）出版的“图与组合学中的矩阵论”和由高等教育出版社出版的“拟阵论”。1996年获学院最优科研奖，2006年获学院最优教师奖和全校最优教师奖，成为西弗吉尼亚大学历史上第一个获此荣誉的华裔教授。曾主持过1996年由美国国家自然科学基金会资助的纪念凯特林（Catlin）教授的欧拉图问题专题会议，2008年由美国国家自然科学基金会资助的第46届美国中西部图论会议，以及2018年的第59届美国中西部图论会议。曾任Discrete Mathematics杂志客座编辑，现担任Applied Mathematics，Graphs and Combinatorics，Congressus Numerantium等多个杂志的编辑。

摘      要：

        Motivated by Thomassen’s conjecture that every 4-connected line graph is Hamiltonian, and as line graphs are $K_{1,3}$-free graphs, many researchers have investigated the Hamiltonian properties of graphs forbidding certain induced graphs including $K_{1,3}$. The hourglass $\Gamma_0$ is the unique simple graph with degree sequence $(4,2,2,2,2)$ and $P_n$ is the path on $n$. In [Discrete Mathematics 341 (2018) 1806-1815],  Z. Ryj\'{a}\v{c}ek, P. Vr\'{a}na and L. Xiong posed a conjecture that every 3-connected $\{K_{1,3}, \Gamma_0, P_{16}\}$-free  graph is Hamilton-connected. X. Liu and L. Xiong  in [Discrete Mathematics 345(2022), 112910] proved this conjecture. We continue to study this problem aiming to characterize all extremal graphs. We have found a family ${\cal W}$ of graphs formed by subdividing the Wagner graph and by attaching pendent vertices, and prove that every 3-connected $\{K_{1,3}, \Gamma_0, P_{18}\}$-free graph $G$ is Hamilton-connected unless the Hamilton-connected closure of $G$ is a member of ${\cal W}$.

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