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

子图扩张与图着色问题 图论是一个古老的，但重新诞生和充满活力的数学分支。自20世纪60年代以来，它迅速发展，其应用扩展到物理学，生物学和运筹学，特别是计算科学和通信网络。图表经常被用作各种科学研究的工作框架。特别是，计算科学为图论的发展提供了许多有趣的问题。 最近，图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。 建议的计划是扩大我们的知识，图的因素，子图的扩展和连接到其他组合的主题。目标是通过以下三种方式获得研究和HQP培训所需的知识： 1)为了理解子图扩展图的结构（即，（Y，H）-可扩展图）的一般递归参数的目的，通过开发一个分解过程。 这样的分解将是至关重要的设计有效的算法来识别和构造的家庭，这样的图。这一工作涉及到对现有技术和方法的概括，以及为通用框架和抽象模型创建新的分析工具。 2)该研究将为子图扩展在其他组合问题和其他数学分支中的潜在应用提供一个更加一致和通用的框架。 概念（Y，H）-可扩展图是一个定义良好的框架，它不仅巩固了许多众所周知的概念（例如，因子临界图、双临界图和缺陷D匹配）在一起，并且还保持其子类的基本性质。这使我们能够简化许多以前的证明，并建立更密切的联系，以其他图论问题。 3)在我们的建议中，我们提出了许多密切相关和明确界定的问题。这些问题有不同程度的困难，从结构和开放的问题，推广已知的结果和特定类别的图形的建设，我们也仔细选择和混合的问题，考虑我们的短期和长期目标的建议。 所提出的问题将实现我在子图扩张研究中的愿景，也为本科生和研究生提供参与创造，探索和体验严谨研究的机会。

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.