Cycle decompositions of graphs and related problems

近20年来，我一直着迷于图形的循环分解。这个充满活力的图论和组合设计理论交叉的研究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，那么这个图就被称为循环。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个参与者是否可以在k个晚上坐在一定数量的指定大小的圆桌旁，以便每对参与者恰好坐在一起一次（假设桌子大小加起来为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。在这个研究计划中，我建议解决这个问题的两个新变体，主要是相等的表大小。我还打算研究完全对称等部有向图的Oberwolfach问题的一个版本，以及从另一个角度研究循环分解问题：使用一种称为合并分离的技术，从现有的完全多部图的循环分解中获得新的完全多部图的循环分解。***我研究计划的第二个主题——超图的欧拉性质——与第一个主题有关，因为连通图允许分解成循环（长度不能预先确定），当且仅当它允许欧拉巡回（每条边正好遇到一次的图的循环遍历）。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，最近已经证明，即使对于超图的某些受限子类，类似问题也是np完全的。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（严格）欧拉巡回的存在性问题也是如此。这两个概念是相关的，都可以用来为某些类型的调度建模。因此，我预计所提出的工作不仅会在这一研究领域留下重要的印记，而且还会找到实际应用。*****

I have been fascinated by cycle decompositions of graphs for nearly 20 years. This vibrant research area on 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 a number of round tables of specified sizes for k 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 wide open. In this research plan, I am proposing to solve two new variants of the problem, primarily for equal table sizes. I also intend to investigate a version of the Oberwolfach Problem for complete symmetric equipartite digraphs, as well as study cycle decomposition problems from another point of view: using a technique called amalgamation-detachment to obtain new cycle decompositions of complete multipartite graphs from existing cycle decompositions of complete multigraphs.*** The second topic of my research proposal – eulerian properties of hypergraphs – is related to the first since a connected graph admits a decomposition into cycles (of lengths that cannot be predetermined) if and only if it admits an Euler tour (cyclic traversal of the graph that encounters each edge exactly once). 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, it has recently been shown that the analogous problem is NP-complete even for some restricted subclasses of hypergraphs. 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 a (strict) Euler tour of a hypergraph. The two concepts are related, and both can be used to model certain types of scheduling. Thus, I anticipate that the proposed work will not only leave a significant mark on this research area, but also find practical applications.*****