Discovery of new graph invariants to capture the cycle ctructure

2022 Fiscal Year Final Research Report

• Project/Area Number: 20K11684
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Review Section: Basic Section 60010:Theory of informatics-related
• Research Institution: Nihon University
• Principal Investigator: SAITO Akira 日本大学, 文理学部, 教授 (90186924)
• Project Period (FY): 2020-04-01 – 2023-03-31
• Keywords: ハミルトンサイクル / サイクル / 次数和 / 辺着色 / 虹色禁止部分グラフ

Outline of Final Research Achievements

If a graph nearly satisfies a sharp sufficient condition for the existence of a hamiltonian cycle, there might be some sign suggesting that it is "almost" hamiltonian. In order to investigate this possibility, we studied the relationship between the degree sum of general graphs and that of its spanning bipartite subgraphs. We proved that a graph G of even order contains a spanning balanced bipartite subgraph H such that the degree sum of H is not less than approximately the half of the degree sum of G. It implies that a extended version of Moon-Moser Theorem is stronger than Ore's Theorem. We also proved the counterpart of this result for graphs of odd order. Next we considered edge-coloring as a generalization of spanning subgraphs. Let H be a connected graph and K be its subgraph. Then every rainbow H-free complete graph edge-colored in sufficiently many colors is rainbow K-free if and only if K is a star and H is its simple subdivision.

Free Research Field

グラフ理論

Academic Significance and Societal Importance of the Research Achievements

ハミルトンサイクルは輸送網における効率的配送、符号の長周期巡回生成など幅広い応用をもつ一方、それ自体が多くの理論的興味を引く研究対象である。しかし一般のグラフにハミルトンサイクルの存在を問う問題はNP-完全問題である。こうした状況において、ハミルトンサイクル発生の予兆を数値的に捉えることは、理論、応用の両面において知見を深める。全域２部グラフの観点からこの問題にアプローチした本研究は、問題に新たな視点を与え、また本研究の成果はこの視点の有効性を示す。また全域部分グラフを辺着色へと一般化する発想も従来なかったものであり、辺着色の分野に新たな研究の場を切り開く。