COLOURING, DOMINATION AND DISCRETE DYNAMIC GRAPH PROCESSES

我的研究计划的原则、目标和长期愿景是探索和推进网络资源的优化配置。短期目标集中在三个领域，着色和分隔，独立性和控制性，以及图表中的动态过程，支持这个项目的长期目标。我的项目包括高素质人才(HQP)的培训，以建立一个多元化和具有竞争力的研究基地。在加拿大提供培训以发展强大的数学文化是至关重要的，这样学生才能拥有成功的技能和知识。色彩的概念通过简单但困难的问题抓住了人们的兴趣。例如，一张地图可以用四种颜色来着色，这样共享边界的国家就会得到不同的颜色吗？这个问题是图论概念中的核心问题。这项研究计划的一个重点是检查地图着色的变化，例如将它们视为图的顶点集的划分，以促进该领域的理论知识。行业，如蜂窝网络，应用着色理论将购买渠道的成本降至最低。着色理论与独立性和支配权密切相关。在这些话题之间找到新的联系是一个长期目标。我们将探索色彩、独立性和支配性之间已知的极端关系。为了帮助开发新的技术，相关的联系将通过将我们的注意力限制在图的一个较小的子类上来改进。离散动态过程被用来对许多具有现实应用程序的迷人游戏进行建模。人们可以将动态控制视为在灾难或紧急情况下部署移动资源中心。这些移动单位的位置和移动方式必须足以对任何一系列紧急情况作出反应。优化资源利用往往是至关重要的。“消防员问题”模拟了火势在地图上的蔓延和控制。我们的目标是确定保护地图一定比例所需的最低资源数量。另一个目标是将火灾在被控制之前烧毁的节点数量降至最低。该模型也适用于通过网络传播的病毒，或通过社交媒体传播的谣言/假新闻。有些令人惊讶的是，动态支配中资源中心数量的界限与色彩和独立性密切相关。消防员问题的答案经常使用与地图着色和通道分配问题相同的技术来找到。该研究计划建立在这些领域已取得的成功的基础上，为探索和推进最佳资源分配做出有意义的贡献。该计划的另一个重要影响是为HQP提供的高质量研究、培训和发展，以支持下一代的未来成功。

The principle objective and long-term vision of my research program is the exploration and advancement of optimal resource allocation in networks. Short-term objectives focusing in three areas, colouring and partitions, independence and domination and dynamic process in graphs, support the long-term aim of this program. Included in my program is the training of highly qualified personnel (HQP) to build a diversified and competitive research base. It is essential to provide the training to develop a strong mathematical culture in Canada so students have the skills and knowledge to succeed. The concept of colouring captures people's interest via simply stated, but difficult questions. For example, can a map be coloured with four colours so that countries sharing a border receive different colours? This question is central in the concept of Graph Theory. A focus of this research program examines variations of map colourings, such as viewing them as a partition of the vertex set of a graph, to advance theoretical knowledge in the field. Industries, such as cellular networks, apply the theory of colouring to minimize the cost of purchasing channels.  The theory of colouring is intimately related to independence and domination. Finding new connections between these topics is a long-term objective. The extremes of known relationships between colouring, independence and domination will be explored. To aid in the development of new techniques, relevant connections will be refined by restricting our attention to a smaller, subclass of graphs. Discrete dynamic processes are used to model many fascinating games that have real-life applications. One can view dynamic domination as deploying mobile resource centers during a disaster or emergency situation. These mobile units must be situated and moved in such a way that they sufficiently respond to any sequence of emergencies. It is often critical to optimize the use of resources. The "firefighter problem" models the spread and containment of fire over a map. Our aim is to determine the minimum number of resources required to protect a certain proportion of the map. A further objective is to minimize the number of nodes a fire burns before being contained. The model can also be applied to a virus spreading through a network, or a rumour/fake news through social media. Somewhat surprisingly, bounds on the number of resource centers in dynamic domination are closely related to colourings and independence. Answers to questions poised in the firefighter problem are often found utilizing the same techniques as the map colouring and channel assignment problem. This research program builds on the established success in these areas to deliver meaningful contributions to the exploration and advancement of optimal resource allocation. An additional important impact of the program is the quality research training and development provided for HQP to support the future success of the next generation.