Decompositions and designs

图是一个数学对象，它可以被可视化为点（顶点）的集合以及连接一些顶点对的线（边）的集合。这个通用的概念可以用来建模许多涉及对象之间的连接或关系的现实生活中的情况。例如，可以生成一个图来对社交网络平台进行建模，其中顶点表示用户，而边表示两个用户之间的友谊。循环是一个图，在该图中，可以从一个顶点开始，沿沿着行进到另一个顶点，并继续沿沿着行进，直到到达开始的顶点（没有其他顶点或边）。图的汉密尔顿圈是包含在该图中的使用所有顶点的圈。一个流行的研究问题是，在一个图中，每一个顶点都至少看到另一半顶点的边（狄拉克图），必须包含多少个汉密尔顿圈。与一小群研究人员一起，我试图在超图的背景下研究这个问题，超图是图的自然概括。我们使用了“随机游走”的创新分析来研究这个问题，在这个问题中，我们想象自己从超图的一个任意顶点开始，并投掷一枚复杂的硬币来决定下一个顶点是什么，沿着一条边移动。这些技术使我们能够证明狄拉克超图确实包含非常多的汉密尔顿圈的新结果（或至少，超图的类似物），实际上，本质上尽可能多.与另一小组研究人员一起，我目前正在研究一个与安徒生猜想有关的问题，该猜想假定任何具有所有可能的边的图都以特殊但自然的方式呈现和着色，必须包含一个“路径”，它使用几乎所有的顶点，是彩虹（有不同的颜色在每个边缘）。为了解决这个问题，我们正在开发“切换”技术，即对一个图形进行小的局部更改，将其变成一个不同的图形，目的是研究有关随机选择这些图形的某些事件的概率。我们还使用新的和令人兴奋的技术，包括把我们的图变成超图，以确保我们在原始图中执行的某些过程在某种意义上“有效”运行。总之，这些技术使我们能够证明新的结果，几乎所有这些特殊颜色的图实际上都包含一个彩虹路径，它使用了图的所有顶点。我们还试图将这些想法应用于一个关于其他特殊颜色图中略有不同的彩虹结构的问题。

A graph is a mathematical object which can be visualised as a collection of dots (vertices) together with a collection of lines (edges) which join some of the pairs of vertices. This versatile notion can be used to model many real-life situations involving connections or relationships between objects. For example, one could produce a graph to model a social networking platform, where vertices represent users, and edges signify friendship between two users.A cycle is a graph in which you can start at one vertex, travel along an edge to another vertex, and keep travelling along edges until you reach the vertex you started with (with no other vertices or edges). A Hamilton cycle of a graph is a cycle contained in that graph which uses all of the vertices. A popular research question asks how many Hamilton cycles must be contained in a graph in which every vertex sees at least half the other vertices in an edge (a Dirac graph). Together with a small group of researchers, I sought to investigate this question in the context of hypergraphs, which are a natural generalisation of graphs. We used innovative analysis of a `random walk' to study this problem, in which we imagine ourselves starting at an arbitrary vertex of the hypergraph, and flipping a complicated coin to decide what vertex to move to next, along an edge. These techniques allowed us to prove the new result that Dirac hypergraphs indeed contain very many Hamilton cycles (or at least, the hypergraph analogue thereof), and actually, essentially as many as possible.Together with another small group of researchers, I am currently investigating a problem related to Andersen's Conjecture, which posits that any graph which has all possible edges present and coloured in a special but natural way, must contain a `path' which uses almost all the vertices and is rainbow (has different colours on every edge). To tackle this problem, we are developing the technique of `switchings', where one makes small local changes to one graph to turn it into a different graph, with the aim to study the probabilities of certain events concerning randomly picking these graphs. We are also using new and exciting techniques involving turning our graphs into hypergraphs in order to ensure that some process we are performing in our original graph runs `efficiently' in some sense. Together, these techniques have allowed us to prove the new result that almost all of these specially coloured graphs actually contain a rainbow path which uses all vertices of the graph. We are also seeking to apply these ideas to a question concerning slightly different rainbow structures inside other specially coloured graphs.