Randomly perturbed graphs: Problems between random and extremal graph theory

在数学中，不同领域之间的思想和方法的交流往往会提供有趣的新见解，并导致该领域的进步。该项目位于离散数学连接随机和极值图论。随机图理论研究的是典型大型图的分析和性质。在极值图论中，由特定约束和最大化强制的罕见刚性事件被广泛研究。在这两者之间，有随机扰动图的模型，它是稀疏随机图和稠密确定图的结合。这种混乱和秩序的相互作用非常有趣的现象，有助于我们理解离散结构的行为。该项目的重点是在这个新出现的模式，这是目前吸引了很多关注的问题。在纯随机和纯确定性设置之间存在广泛的插值可能性，这为新的挑战性设置提供了空间。大部分的问题都要求嵌入的生成结构，如汉密尔顿圈，因子的图，树，或一般的有界度图。这些问题可以使用两个领域的方法与针对环境的其他新技术相结合来解决。最终，这种扰动模型的进步也有望在纯设置中的开放问题上取得进展。

In mathematics, the exchange of ideas and methods between different areas often provides interesting new insights and leads to progress in the field. This project is located in discrete mathematics connecting random and extremal graph theory. Random graph theory is concerned with the analysis and properties of typical large graphs. In extremal graph theory rare rigid events forced by specific constraints and maximisation are studied extensively. In between these two, there is the model of randomly perturbed graphs, which is the union of a sparse random and a dense deterministic graph. This fascinating interplay of chaos and order features very interesting phenomena and contributes to our understanding of the behaviour of discrete structures. The focus of this project is on problems in this newly emerged model, which is currently attracting a lot of attention. There is a wide range of possibilities for interpolation between the purely random and the purely deterministic setup, which provides room for new challenging conjectures. Most of the questions ask for embeddings of spanning structures, such as Hamilton cycles, factors of graphs, trees, or general bounded degree graphs. These problems can be approached using a combination of methods from both areas with additional new techniques tailored towards the environment. Ultimately, advances in this perturbed model also promise progress in open questions in the pure setups.