Disjoint perfect matchings and disjoint Hamilton cycles of regular graphs
数学学科创建110周年系列报告
报告题目(Title)：Disjoint perfect matchings and disjoint Hamilton cycles of regular graphs
报告人(Speaker)：晏卫根教授（集美大学）
地点(Place)：腾讯会议 ID：823 670 238 密码：287367
时间(Time)：2025年11月28日（周五）10:30-11:30
邀请人(Inviter)：徐敏
报告摘要
Let G be a connected k-regular graph of order n, and let p(G) and h(G) denote the number of disjoint perfect matchings and disjoint Hamilton cycles of G, respectively. The 1-factorization (resp., Hamilton decomposition) conjecture states that p(G) = k if n ≤ 2k and n is even (resp., h(G) = k/2 if n ≤ 2k+1). Zhang and Zhu (J. Combin. Theory Ser. B, 56 (1992) 74-89) proved that p(G) ≥ k/2 if n ≤ 2k and n is even, and Jackson (J. London Math. Soc. 19 (1979) 13-16) proved that h(G) ≥ (3k-n+1)/6 if 14 ≤ n ≤ 2k+1.
In this paper, we mainly obtain the following two results, which extend the above results by Zhang and Zhu and by Jackson.
1.If n ≤ 3k+3 and n is even, then p(G) ≥ min{λ₀(G), k-⌈n/4⌉}, where λ₀(G) is the odd-edge-connectivity of G.
2.If G is 2-connected and n ≤ 3k, then h(G) ≥ min{λ(G) -⌊k/2⌋, ⌊(3k-n+1)/6⌋}, where λ(G) is the edge-connectivity of G.
As an application, we prove that if a k-regular graph G satisfies: λ₁(G) ≥ k-2, 3n ≤ 8k, k is odd, and k=k₁ + k₂ + ⋯ + kₘ is any partition of k with each part kᵢ ≥ 2, then G has a (G₁,G₂,...,Gₘ)-factorization, where λ₁(G) is the size of smallest odd edge-cut of G, and Gᵢ is a kᵢ-factor of G for I = 1,2,...,m, which partially confirms a conjecture posed by Thomassen (J. Combin. Theory Ser. B, 141 (2020) 343-351).
This is joint work with Jingchao Lai and Xing Feng.
主讲人简介
晏卫根，集美大学教授、博士生导师。2003年7月获厦门大学理学博士学位，2004年10月至2006年12月在中国台湾“中研院”从事博士后研究工作。主要从事组合与图论及其在统计物理中的应用方面的研究工作，在Journal of Combinatorial Theory Ser. A、Journal of Graph Theory与Advances in Applied Mathematics等国际期刊上发表学术论文90多篇。曾获福建省自然科学一等奖，已获多项国家自然科学基金面上项目的支持。
