Subgraph extension problem: structures, characterizations and its connection with edge-weighting coloring problems

子图扩展问题：结构、表征及其与边加权着色问题的联系

• 批准号: RGPIN-2014-05317
• 负责人: Yu, Qinglin
• 金额: $ 0.8万
• 依托单位: Thompson Rivers University
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2018
• 资助国家: 加拿大
• 起止时间: 2018-01-01 至 2019-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=669448
• 关键词: Subgraph; extension; problem; structures; characterizations

Subgraph extension and graph coloring problems* Graph Theory is an old but re-born and energized branch of mathematics. It has grown rapidly since the 1960's with its applications spreading to Physics, Biology and Operations Research, in particular to Computing Science and communication networks. Often graphs are used as a working frame for various scientific investigations. In particular, Computing Science has provided many interesting problems for Graph Theory to grow. More recently, Graph Theory has become a useful instrument for the study of gene sequences, environment sustainability, management science and logic designing. * The proposed program is to expand our knowledge on graph factors, subgraph extension and its connection to other combinatorial topics. The objectives are to acquire knowledge, both for research and HQP training in three ways:* 1) To understand the structures of subgraph extension graphs (i.e., (Y, H)-extendable graphs) by developing a decomposition procedure for the purpose of general recursive arguments. Such a decomposition will be vital for the design of efficient algorithms to recognize and to construct the family of such graphs. This work involves to generalize the existing techniques and methods, and to create new analysis tools for the general framework and abstract models. * 2) The research will deliver a more consistent and universal framework for the potential applications of subgraph extension to other combinatorial problems and other mathematical branches. The concept, (Y, H)-extendable graphs, is a well-defined framework, which not only consolidate many well-known concepts (e.g., factor-critical graph, bicritical graphs and defect-d matching) together and also maintain the basic properties of its sub-classes. This enables us to simplify many of the previous proofs and establish closer connections to other graph theory problems. * 3) In our proposal, we have stated many closely related and well-defined problems. These problems have different levels of difficulties, from conjectures and open problems, to generalization of known results and construction of specified classes of graphs; we also carefully select and blend the problems in proposal by considering our short-term and long-term objectives. The problems proposed will fulfill my vision in the study of subgraph extension and also provide opportunities for involvement of undergraduate and graduate students to engage in creation, exploration and experiencing rigorous research.

子图扩展和图形着色问题*图理论是数学的旧但重生且充满活力的分支。自1960年代以来，它的应用已扩展到物理，生物学和运营研究，尤其是计算科学和通信网络。通常，图被用作各种科学研究的工作框架。特别是，计算科学为图理论提供了许多有趣的问题。 最近，图理论已成为研究基因序列，环境可持续性，管理科学和逻辑设计的有用工具。 *拟议的程序是扩展我们对图形因素，子图扩展及其与其他组合主题的联系的知识。这些目标是通过三种方式获取研究和HQP培训的知识：* 1）通过为一般递归参数的目的制定分解程序，以了解子图扩展图（即（y，h） - 可扩展图）的结构。 这样的分解对于设计有效算法的设计和构建此类图的家族至关重要。这项工作涉及概括现有技术和方法，并为一般框架和摘要模型创建新的分析工具。 * 2）该研究将为子图扩展到其他组合问题和其他数学分支的潜在应用提供更加一致和普遍的框架。 该概念（y，h） - 可扩展图是一个明确定义的框架，它不仅合并了许多众所周知的概念（例如，关键因素图，两性图形图和缺陷d匹配），还保持其子类的基本属性。这使我们能够简化以前的许多证据，并与其他图理论问题建立更紧密的联系。 * 3）在我们的提案中，我们指出了许多密切相关且定义明确的问题。这些问题有不同程度的困难，从猜想和开放问题到已知结果的概括以及指定图类别的构建；我们还通过考虑我们的短期和长期目标来仔细选择并融合提案中的问题。 提出的问题将满足我在研究子图扩展方面的愿景，还为本科生和研究生参与参与创建，探索和经历严格的研究提供了机会。