Topological Graph Theory

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Purpose of Research

Topological Graph Theory can be generally defined as the mathematics used to explore the structure and phenomena of graphs embedded on closed surfaces. Until Four Color Problem and Map Color Theorem were solved, finding their solutions was a central issue in this field. However, in the 1980s, Negami and his research group created many new research themes and pioneering a new era in topological graph theory. Negami held “Workshop on Topological Graph Theory” every year since he arrived at Yokohama National University. The name of this workshop became abbreviated to TGT followed by the year in Heisei format and became widely recognized around the world. In particular, TGT10, TGT20, TGT25 and TGT30 were held as international conferences and grew to more than 100 participants from Japan and abroad.

As shown in the attached figure, the research theme is diverse, but many of them can be identified along with the keywords “re-embedding of graph.” Thus, in recent years, we have gained Grants-in-Aid for Scientific Research for the central theme of “graph re-embedding structures,” and we have thereby operated the research center. Meanwhile, Konno, who participates as a new member in this application, forms a community to vigorously study the theory of quantum walks. In recent years, the collaboration between Negami and Konno has resulted in the development of a method to analyze quantum walks on a graph, in relation to theory of covering spaces of graphs. Therefore, we aim to integrate the results of research from this research center with probability theory, to develop a new field in topological graph theory and plan a smooth generation change.


Research content / methods

In this research, rather than limiting the topic to a single area, we will explore several research topics that can be referenced within the framework of Topological Graph Theory, and we will build a comprehensive knowledge system. To that end, we will secure a system that allows discussions at various levels. The minimum unit is a seminar in each laboratory, but in “Seminar on Topological Graph Theory” led by Nakamoto, graduate students and researchers will gather by crossing the boundaries of laboratories and universities, identify problems that can be solved in relation to the graph re-embedding structures, and they will attempt to solve them. Further, in November each year, the aforementioned “Workshop on Topological Graph Theory” shall be held to bring together a wide variety of researchers related to this research, publish the results of research, and discuss them. In particular, TGT28 was an academic gathering commemorating Negami’s 60th anniversary, while TGT30 was held as an international conference. In addition, we will develop international joint research work utilizing the networks that we have built thus far with researchers from the US, UK,New Zealand, Slovakia, Czech Republic, Mexico, Spain, and South Korea, etc.

In addition to the purely mathematical discussions, we will attempt to utilize computers to automatically create irreducible triangulations and their panel structures on closed surfaces, and to develop a method for managing big data generated through this process. This makes it possible to observe phenomena that were otherwise not seen with manual work only. We expect that a new world of Topological Graph Theory will be revealed, and we would be providing a new place where the next generation of researchers can work.