Cycle decompositions of graphs and eulerian properties of hypergraphs

20多年来，我一直着迷于图形的循环分解。这个充满活力的研究领域是图论和组合设计理论的交叉领域，在过去几年中取得了惊人的进展。尽管如此，一些核心问题仍然悬而未决，而且由于最近的进展，现在已经更容易解决了。我的长期愿景是为这一知识体系做出重大而持久的贡献。如果一个图的边可以上色，使得任意一种颜色的所有边的集合形成一个循环，那么这个图就被分解成循环。中心循环分解问题之一是Oberwolfach问题，于1967年首次在调度的背景下被引入。它问一个会议的n个参与者是否可以一连几个晚上坐在指定大小的圆桌旁，以便每对参与者恰好挨着坐一次（假设桌子大小加起来等于n）。用数学术语来说，这个问题是问一个完全图是否可以分解成2个因子（夜），每个因子由特定长度的循环（表）组成。对于这个问题的原始版本，已知有许多针对特定表大小的解决方案，但是，这个问题通常仍然是开放的。在这个应用程序中，我建议研究这个问题的几个变体：爱人型、蜜月型和定向型。前两个变体涉及n/2对；每个参与者都要恰好坐在其他参与者旁边一次，恰好坐在配偶旁边两次（爱配偶的变体），或者每次都坐在配偶旁边（蜜月变体）。在有向变量中，每个参与者只坐在其他参与者的右边一次。此外，我建议从另一个角度研究循环分解问题：使用一种强大的技术，分离，从已有的完全多部图的循环分解中得到新的完全多部图的循环分解。我的研究计划的第二个主题是超图的欧拉性质（它是图的一种概括）。众所周知，当且仅当每个顶点都有偶度时，图才允许欧拉游。超图没有类似的结果；事实上，超图的类似问题是np完全的（计算困难）。此外，有不止一种自然的方法可以将欧拉游的概念推广到超图，我建议根据这些性质研究各种超图。给定图分解成特定长度的圈的存在性问题是图论中的一个基本开放问题，超图的欧拉巡回的存在性问题也是如此。这两个概念是松散相关的，并且都可以用于为某些类型的调度问题建模。因此，所提出的工作不仅将在这些研究领域留下重要的印记，而且具有实际应用价值。

I have been fascinated by cycle decompositions of graphs for over 20 years. This vibrant research area in the intersection of graph theory and the theory of combinatorial designs has made breathtaking progress in the last few years. Nevertheless, some central problems remain open and have now become more accessible because of the recent advances. It is my long-term vision to make a significant and lasting contribution to this body of knowledge. A graph is said to be decomposed into cycles if its edges can be coloured so that the collection of all edges of any one colour forms a cycle. One of the central cycle decomposition problems, first introduced in 1967 in the context of scheduling, is the Oberwolfach Problem. It asks whether n participants at a conference can be seated at round tables of specified sizes for several nights so that each pair of participants sit next to each other exactly once (assuming that the table sizes add up to n). In mathematical terms, the problem asks whether a complete graph can be decomposed into 2-factors (nights), each consisting of cycles (tables) of specified lengths. For this original version of the problem, many solutions are known for specific table sizes, however, the problem is in general still open. In this application, I am proposing to investigate several variations of the problem: the spouse-loving variant, the honeymoon variant, and the directed variant. The first two variants involve n/2 couples; each participant is to sit next to every other participant exactly once, and next to their spouse exactly twice (the spouse-loving variant) or every time (the honeymoon variant). In the directed variant, each participant is to sit to the right of every other participant exactly once. In addition, I propose to study cycle decomposition problems from another point of view: using a powerful technique, detachment, to obtain new cycle decompositions of complete multipartite graphs from existing cycle decompositions of complete multigraphs. The second topic of my research proposal is eulerian properties of hypergraphs (which are a generalization of graphs). It is well known that a graph admits an Euler tour if and only if every vertex has even degree. No similar results are known for hypergraphs; in fact, the analogous problem for hypergraphs is NP-complete (computationally hard). Moreover, there is more than one natural way to generalize the notion of an Euler tour to hypergraphs, and I propose to investigate various classes of hypergraphs with respect to these properties. The question of existence of a decomposition of a given graph into cycles of specified lengths is a fundamental open problem in graph theory, as is the question of existence of an Euler tour of a hypergraph. The two concepts are loosely related, and both can be used to model certain types of scheduling problems. Thus, the proposed work will not only leave a significant mark on these research areas, but also has practical applications.