一致超图Hamilton圈的存在性条件研究

The study of the existence of the Hamilton cycle has always been a core and fruitful field in graph theory. The development of hypergraph regular lemma and absorbing lemma greatly promotes the study of the hypergraph Hamilton cycle theory. This project mainly study the Ore-type condition of the Hamilton cycle in uniform hypergraphs by using the hypergraph regularity lemma, absorption lemma and probability method. The specific research contents of this project are as follows: (1) In the case of l≤r-2, the approximate Ore-type condition of the Hamilton l-cycle with respect to the minimum (r-1)-degree; (2) The exact Ore type results of Hamilton l-cycle with respect to the minimum (r-1)-degree; (3) The approximate Ore type condition of the Hamilton tight cycle with respect to the minimum d-degree in the case of d ≤ r-2. This project systematically studies the Ore-type conditions of the Hamiltonian cycle in uniform hypergraphs, which would enrich the understanding of the hypergraph structure and the results of the Ore type condition of hypergraph Hamilton cycle.

图中Hamilton圈存在性的研究一直是图论中的一个核心和富有成果的领域。近年来，超图正则引理和吸收引理的出现极大地推动了超图Hamilton圈理论的研究。本项目主要利用超图正则引理，吸收引理，概率方法等来研究一致超图中Hamilton圈的Ore型条件。本项目的具体研究内容如下：(1) l≤r-2情形下，Hamilton l-圈关于最小（r-1）度的近似Ore型条件；(2) Hamilton l-圈关于最小（r-1）度的精确Ore型条件；(3) d≤r-2情形下Hamilton紧圈关于最小d度的近似Ore型条件。本项目系统地对超图Hamilton圈的Ore型条件进行深入研究，丰富对超图结构的认识及Hamilton圈Ore型条件的结果。

Hamilton圈的存在性研究一直是图论中的一个核心和富有成果的领域。近年来，超图正则引理和吸收引理的出现极大地推动了超图Hamilton圈理论的研究。本项目主要利用超图正则引理，吸收引理，概率方法等研究了一致超图中Hamilton圈的存在性问题及相关问题。具体研究成果如下：1. 得到了k一致超图系统中彩虹Hamilton紧圈的最小k-2度条件，该结果解决了Hamann, Müyesser, Parczyk和Sgueglia最近提出的一个有挑战性的问题；2. 得到了若干超图的anti-Ramsey数的精确值，包括线性路和圈，松路与松圈，不交星图的扩张以及第二边临界图的扩张，该结果完善并推广了Gu, Li和Shi以及Jiang和Pikhurko的结果。3. 得到了扇图的扩张超图的精确Turán数，该结果推广了Pikhurko关于单个团的扩张超图的Turán数结果。三年内共发表（含在线发表）论文4篇，其中SCI论文4篇。