Guangzhou Discrete Mathematics Seminar

Talks in 2022

2022.3

21 December 2022 (Wednesday), 7:30-8:30pm Beijing time, online, Tencent Meeting ID: 184 018 751

Fang Tian (Shanghai University of Finance and Economics, Shanghai, China) Enumeration of linear hypergraphs with given size and its applications Poster Slides

For n ≥ 3, let r = r(n) ≥ 3 be an integer. A hypergraph is r-uniform if each edge is a set of r vertices, and is said to be linear if two edges intersect in at most one vertex. In this talk, the number of linear r-uniform hypergraphs on n → ∞ vertices is determined asymptotically when the number of edges is m(n) = o(r⁻³n^3/2). We also find the probability of linearity for the independent-edge model of random r-uniform hypergraph when the expected number of edges is o(r⁻³n^3/2); and some recent developments as the applications.

2022.2

20 June 2022 (Monday), 4-5:30pm Beijing time, online, Tencent Meeting ID: 770 491 425

Yubao Guo (RWTH Aachen University, Germany) Extensions of Alspach's theorem to regular multipartite tournaments Poster Slides

An arc of a digraph D is called pancyclic, if it lies on a cycle of length t for all t ∈ {3, ... , |V(D)|}. Alspach (On cycles of each length in regular tournaments, Canad. Math. Bull. 10, 1967) proved that every regular tournament T is arc-pancyclic, i.e., each arc of T is pancyclic.
Since multipartite tournaments, although a natural generalization of tournaments, don’t have the same arc-pancyclicity as tournaments, we have tried to extend the classical cycle concept to multipartite tournaments in various ways.
In this talk, we will give an overview of quasi_x-arc-pancyclicity, x ∈ {p, l, o, nl, ps}, arc-pandashcyclicity in regular multipartite tournaments and leave a few open problems on this topic.

2022.1

20 May 2022 (Friday), 11am-12pm Beijing time, online, Tencent Meeting ID: 184 018 751

Bo Ning (Nankai University, Tianjin, China) Rainbow triangles in edge-colored graphs Poster Slides

In this talk, we shall survey our work on rainbow triangles in edge-colored graphs. In particular, we will give a sketch of a recent theorem of ours. This counting result states that the number of rainbow triangles in an edge-colored graph G is at least δ^c(G)(2δ^c(G) − n)n/6, which is best possible by considering the rainbow k-partite Turán graph, where its order is divisible by k. This means that there are Ω(n²) rainbow triangles in G if δ^c(G) ≥ (n + 1)/2, and Ω(n³) rainbow triangles in G if δ^c(G) ≥ cn when c > 1/2. This can be seen as a counting version of a previous theorem due to Hao Li.

References
[1] B. Li, B. Ning, C. Xu, S. Zhang, Rainbow triangles in edge-colored graphs, European J. Combin. 36 (2014), 453-459.
[2] S. Fujita, B. Ning, C. Xu, S. Zhang, On sufficient conditions for rainbow cycles in edge-colored graphs, Discrete Math. 342(7) (2019), 1956-1965.
[3] X. Li, B. Ning, Y. Shi, S. Zhang, Counting rainbow triangles in edge-colored graphs, arXiv:2112.14458.