题目一：Cycles in multipartite tournaments(多部竞赛图的圈性质)
内容简介：In this talk, I will introduce our recent work on cycles in multipartite tournaments. Classical results on cycles in tournaments states that a tournament with a Hamiltonian cycle is also pancyclic and cycle extendable, which require that every nonhamiltonian cycle can be extended to another cycle with exactly one more additional vertex. We pursue similar results for multipartite tournaments. We find counterexamples for multipartite tournaments with at least 3 partite sets, and for bipartite tournaments, we prove that Hamiltonicity, pancyclicity and cycle extendability are equivalent, with one class of exceptional digraphs, which can be clearly characterized. As an important tool in our work, we define an auxiliary graph called the in-out graph of a digraph, which can be viewed as a generalization of the concept of line graphs to digraphs.
报告人：广东轻工职业技术学院 张赞波 教授
报告人简介：广东省“千百十”人才培养工程省级培养对象。先后在中山大学（2008.6）和荷兰特文特大学（University of Twente，2017.9）获得计算机和应用数学方向博士学位。主要从事图论及其算法等方面研究工作。在SIAM Journal of Discrete Mathematics，中国科学等重要国际国内学术期刊上发表论文近二十篇，完成学术著作两本。在图的匹配理论，以及路与圈理论的方向上取得一些基础性的成果，被相关领域的专著和综述所引用。主持广东省自然科学基金项目两项，以第一参与人参加国家自然科学基金项目一项。
题目二：Minimum number of edges in odd cycles
内容简介：If a graph $G$ has $n/ge4k$ vertices and more than $n^2/4$ edges, then it contains a copy of $C_{2k+1}$. In 1992, Erd/H{o}s, Faudree and Rousseau showed even more, that the number of edges that occur in a triangle of such a $G$ is at least $2/lfloor n/2/rfloor+1$, and this bound is tight. They also showed that the minimum number of edges that occur in a $C_{2k+1}$ for $k/ge2$ is at least $11n^2/144-O(n)$, and conjectured that for any $k/ge2$, the correct lower bound should be $2n^2/9-O(n)$. Very recently, F/"uredi and Maleki constructed $n$-vertex graphs with more than $n^2/4$ edges such that only $(2+/sqrt{2}+o(1))n^2/16/approx0.213n^2$ of them occur in $C_5$, which disproves the conjecture for $k=2$. They also proved that this construction is asymptotically best possible by showing that for any $/varepsilon>0$, graphs with $(1/4+/varepsilon)n^2$ edges contain at least $(2+/sqrt{2}-/varepsilon)n^2/16$ edges that occur in $C_5$. In this paper, we use a different approach to tackle this problem and obtain the following stronger result: Any $n$-vertex graph with at least $/lfloor n^2/4/rfloor+1$ edges has at least $(2+/sqrt{2}-o(1))n^2/16$ edges that occur in $C_5$, which answers a conjecture of F/"uredi and Maleki. For $k/ge3$, we adapt the approach to show that $n$-vertex graphs with more than $n^2/4$ edges contain at least $2n^2/9-o(n^2)$ edges that occur in $C_{2k+1}$. For both of these results, we also prove stability of the corresponding extremal constructions.Join work with Andrzej Grzesik and Jan Volec.
报告人：中山大学 胡平 副教授
报告人简介：一直从事图论领域中极值图论方向的研究，在领域内的Ramsey理论，Turán理论，染色问题等方向均有成果。现主要研究方向之一为应用和拓展芝加哥大学Razborov教授创建的旗代数(Flag Algebra)理论。此理论已被用于解决多个有数十年历史的猜想。胡平应用此理论与合作者解决了Erdős和Sós在1972年提出的彩虹(rainbow)三角形最大个数的猜想，以及Pippinger和Golumbic在1975年提出的C5的最大导出密度(induced density)的猜想，还解决了Erdős, Faudree和Rousseau在1992年提出的奇圈中边数的猜想。目前其论文均发表在图论领域的国际一流期刊上。
时 间：2018年10月22日（周一）下午2：30始
地 点：南海楼224室
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信息科学技术学院/网络空间安全学院
2018年10月18日