学术报告（ Boram Park， Shinya Fujita 1.7）
Boram Park (韩国Ajou University)
Unavoidable subgraphs in a graph with large matching number
Given a graph parameter ρ, every graph G with sufficiently large ρ(G) contains a ‘well-structured’ induced subgraph H with large ρ(H). The classical Ramsey’s theorem deals with the case when the graph parameter under consideration is the number of vertices; there is also a Ramsey-type theorem regarding connected graphs. In other words, Ramsey’s theorem is for unavoidable structures in a graph with large number of vertices.
Given a graph G, the matching number and the induced matching number of G is the maximum size of a matching and an induced matching, respectively, of G. In this paper, we formulate Ramsey-type theorems for the matching number and the induced matching number regarding connected graphs. Along the way, we obtain a Ramsey-type theorem for the independence number regarding connected graphs as well. The work is based on joint work with Ilkyoo Choi, Michitaka Furuya, and Ringi Kim.
Shinya Fujita (日本 Yokohama City University)
Some recent results on edge-colored graphs
An edge-colored graph is called properly colored if no two adjacent edges have the same color. Also, an edge-colored graph is called rainbow if no color repeats on it. Nowadays, properly colored cycles or rainbow cycles in edge-colored graphs are widely studied in graph theory. In this talk, some recent results on properly colored cycles and rainbow cycles will be reviewed. I will also present some open problems in this topic.