題目一：Cycles in multipartite tournaments(多部競賽圖的圈性質)
内容簡介：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.
報告人：廣東輕工職業技術學院 張贊波 教授
報告人簡介：廣東省“千百十”人才培養工程省級培養對象。先後在中山大學（2008.6）和荷蘭特文特大學（University of Twente，2017.9）獲得計算機和應用數學方向博士學位。主要從事圖論及其算法等方面研究工作。在SIAM Journal of Discrete Mathematics，中國科學等重要國際國内學術期刊上發表論文近二十篇，完成學術著作兩本。在圖的匹配理論，以及路與圈理論的方向上取得一些基礎性的成果，被相關領域的專著和綜述所引用。主持廣東省自然科學基金項目兩項，以第一參與人參加國家自然科學基金項目一項。
題目二：Minimum number of edges in odd cycles
内容簡介：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.
報告人：中山大學 胡平 副教授
報告人簡介：一直從事圖論領域中極值圖論方向的研究，在領域内的Ramsey理論，Turán理論，染色問題等方向均有成果。現主要研究方向之一為應用和拓展芝加哥大學Razborov教授創建的旗代數(Flag Algebra)理論。此理論已被用于解決多個有數十年曆史的猜想。胡平應用此理論與合作者解決了Erdős和Sós在1972年提出的彩虹(rainbow)三角形最大個數的猜想，以及Pippinger和Golumbic在1975年提出的C5的最大導出密度(induced density)的猜想，還解決了Erdős, Faudree和Rousseau在1992年提出的奇圈中邊數的猜想。目前其論文均發表在圖論領域的國際一流期刊上。
時 間：2018年10月22日（周一）下午2：30始
地 點：南海樓224室
熱烈歡迎廣大師生參加!
太阳集团1088vip/網絡空間安全學院
2018年10月18日