On Factor problems in graph on surfaces

2020 Fiscal Year Final Research Report

• Project/Area Number: 17K05349
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Foundations of mathematics/Applied mathematics
• Research Institution: Keio University
• Principal Investigator: Fujisawa Jun 慶應義塾大学, 商学部(日吉), 教授 (00516099)
• Project Period (FY): 2017-04-01 – 2021-03-31
• Keywords: 位相幾何学的グラフ理論 / 因子問題 / 完全マッチング / 三角形分割

Outline of Final Research Achievements

The following is the main part of the results obtained in this research. Firstly, it turned out that every 3-conncted 3-regular bipartite graph on a surface is distance matchable. Secondly, the problem concerning separating 3-cycles in non-hamiltonian 1-tough triangulation of the plane, posed by Ozeki and Zamfirescu, was solved in the affirmative. Thirdly, as for the existence of the perfect matchings in graphs obtained from 5-connected triangulation of a surface by deleting some vertices, we obtained a generalization of the theorem shown by Kawarabayashi, Plummer and Ozeki. Moreover, we obtaind a short proof and a generalization of the theorem shown by Aldred, Kawarabayashi and Plummer.

Free Research Field

グラフ理論

Academic Significance and Societal Importance of the Research Achievements

本研究では1993年にBroersmaが提起したハミルトンサイクルに関する予想、2018年にOzeki-Zamfirescuが提起した平面の三角形分割に関する問題の2つの未解決問題がいずれも肯定的に解決されたため、その学術的な意義は大きい。また、閉曲面上の5-連結グラフのマッチング拡張問題においてはいくつかの既存の結果が一般化されるとともに、これまでに三角形分割でしか得られていなかった結果を三角形分割でないグラフへと拡張することに成功し、得られる知見が格段に広がった。