报告题目：Refined matroid covering and packing
theorems and their graph applications
报告人：扬大庆教授 浙江师范大学
报告时间：2021.6.3 10:30-11:30
地点：南山校区6号楼407
报告摘要：Let M be a loopless matroid on E with rank function r_M. Let $\Gamma(M)=\max_{\emptyset\neqX\subseteq E}\frac{|X|}{r_M(X)}$ And$\Phi(M)=\min_{r_M(X)<r(M)} \frac{|E|-|X|}{r_M(E)-r_M(X)}$. The Matroid Covering and Packing Theorems state that the minimum number of independent sets whose union is $E$ is $\lceil\Gamma(M)\rceil$, and the maximum number of disjointbases is $\lfloor\Phi(M)\rfloor$.We prove refinements of these theorems. If $\Gamma(M)=k+\varepsilon$, where $k \ge 0$ is an integer and $0 \le \varepsilon< 1$, then $E$ can bepartitioned into $k+1$ independent sets with one of size at most$\varepsilon \cdotr_M(E)$. If $\Phi(M)=k+\varepsilon$, then $M$ has $k+1$ disjoint independent sets such that $k$ are bases and the other has size at least $\varepsilon \cdotr_M(E)$. We apply these results to truncations of cycle matroids to refinegraph-theoretic results due to Chen, Koh, and Peng in 1993%On the higher-order edge toughness of a graph, Discr. Math. 111 (1993) 113-123and to Chen and Lai in 1996.
个人简介：杨大庆，浙江师范大学教授，博士生导师。杨大庆教授主要从事图论中的图分解、图染色、以及图上的算法研究。已经主持完成国家自然科学基金面上项目二项。参与完成了科技部“973计划课题”、国家自然科学基金重点项目，各一项。现主持国家自然科学基金面上项目（“图的广义森林体系及其应用推广研究”）。研究成果包括首先与Kierstead教授一起提出并研究了图的广义染色数概念；与蒋宏弼解决了由朱绪鼎教授等提出的关于图的森林分解的“九龙树猜想”。学术成果发表在《Journal of Combinatorial Theory, Series B》，《Combinatorica》，《European Journal of Combinatorics》等图论与组合领域的重要学术期刊上。
