Integrated study of generating theorem and local deformation theory of graphs on closed surfaces

2023 Fiscal Year Final Research Report

• Project/Area Number: 20K03714
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 12030:Basic mathematics-related
• Research Institution: Niigata University
• Principal Investigator: Suzuki Yusuke 新潟大学, 自然科学系, 教授 (10390402)
• Project Period (FY): 2020-04-01 – 2024-03-31
• Keywords: グラフ / 生成定理 / 局所変形 / 多面体的 / 四角形分割 / 偶三角形分割

Outline of Final Research Achievements

Starting with Wagner's result in the 1930s, research on local transformations of graphs embedded on closed surfaces has developed in various directions. On the other hand, the generating theorem of graphs is a powerful tool for proving propositions by induction, as in the proof of the Four-Color Theorem, but it is also an independent research topic in its own right. We have obtained some results for the two research topics above, and for problems that lie on the boundary between them. In particular, for polyhedral quadrangulations of closed surfaces, we have succeeded in identifying eight reductional operations to reduce such graphs to a finite number of irreducible graphs. We also proved that any two polyhedral quadrangulations can be transformed into each other by a sequence of local transformations called cubical flips, which vary the number of vertices of graphs.

Free Research Field

位相幾何学的グラフ理論

Academic Significance and Societal Importance of the Research Achievements

組合せトポロジー分野でも，Alexander (1930) による基礎的な結果を出発点に多くの結果が存在している．今回，組合せトポロジーと（我々の）位相幾何学的グラフ理論が接触し，その境界に位置する問題に対する研究成果を出したことは大変意義深い．当該研究は我々が長年蓄積してきた局所変形問題と生成定理，それぞれの結果から得られる知見やグラフのリストを用いて行ったものであり，学術的独自性を持ち価値がある．グラフの変形に関する研究は，離散幾何や計算機科学分野でも盛んにおこなわれており，そこでの問題創出を始め，その進展にも影響を及ぼすことが期待できる．