Cycle decompositions of graphs and related problems

图的循环分解及相关问题

• 批准号: RGPIN-2016-04798
• 负责人: Sajna, Mateja
• 金额: $ 1.6万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2016
• 资助国家: 加拿大
• 起止时间: 2016-01-01 至 2017-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=615309
• 关键词: Cycle; decompositions; graphs; related; problems

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.

近20年来，我一直着迷于图的循环分解。这个充满活力的研究领域的交叉图论和组合设计理论取得了惊人的进展，在过去的几年里。然而，一些核心问题仍然悬而未决，而且由于最近取得的进展，这些问题现在变得更加容易解决。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着色，使得任何一种颜色的所有边的集合形成一个圈，则称该图被分解为圈。在1967年首次引入调度的背景下，中央循环分解问题之一是Oberwolfach问题。它询问会议中的n个参与者是否可以在k个晚上坐在指定大小的圆桌上，以便每对参与者恰好坐在一起一次（假设桌子大小加起来是n）。在数学术语中，这个问题是问一个完整的图是否可以分解成2个因子（nights），每个因子由指定长度的循环（表）组成。 对于这个问题的原始版本，许多解决方案是已知的特定表大小，然而，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，主要是针对相等的桌子大小。我还打算调查一个版本的Oberwolfach问题的完全对称等部有向图，以及研究循环分解问题从另一个角度来看：使用一种技术称为汞齐分离，以获得新的循环分解的完全多部图从现有的循环分解的完全多重图。 我的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之旅（循环遍历图，遇到每条边正好一次）。众所周知，一个图允许一个欧拉环当且仅当每个顶点都是偶数度。没有类似的结果是已知的超图;事实上，它最近已被证明，类似的问题是NP完全的，甚至对某些限制的超图子类。此外，有一个以上的自然的方式来推广的概念，欧拉巡回超图，我建议调查各种超图类的这些属性。 一个给定的图分解成指定长度的圈的存在性问题是图论中一个基本的开放问题，就像超图的（严格）欧拉之旅的存在性问题一样。这两个概念是相关的，都可以用来模拟某些类型的调度。因此，我期望所提出的工作不仅会在这一研究领域留下重要的印记，而且会找到实际应用。