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对涉及对象之间的连接或关系的许多现实情况进行建模。例如，一个人可以生成一个图来为社交网络平台建模，其中顶点代表用户，边代表两个用户之间的友谊。循环是一个图，在这个图中，你可以从一个顶点开始，沿着一条边移动到另一个顶点，然后沿着边继续移动，直到你开始时的顶点(没有其他顶点或边)。图的哈密尔顿圈是包含在使用所有顶点的图中的圈。一个流行的研究问题是，在一个图(狄拉克图)中，每个顶点至少能看到一条边中其他顶点的一半，那么图中必须包含多少个哈密尔顿圈。我和一小群研究人员一起，试图在超图的背景下研究这个问题，超图是图的自然推广。我们用“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任意一个顶点开始，然后抛出一个复杂的硬币，以决定下一个顶点将沿着一条边移动到哪个顶点。这些技术使我们能够证明新的结果，即狄拉克超图确实包含非常多的哈密尔顿圈(或者至少，与其类似的超图)，并且实际上，实际上包含尽可能多的哈密尔顿圈。与另一小组研究人员一起，我目前正在研究与安徒生猜想有关的一个问题，该猜想假设任何具有所有可能的边并以特殊但自然的方式着色的图一定包含一条使用几乎所有顶点的彩虹(每条边都有不同的颜色)。为了解决这个问题，我们正在开发“切换”技术，即对一个图形进行微小的局部更改，将其转换为不同的图形，目的是研究与随机挑选这些图形有关的某些事件的概率。我们还使用了新的令人兴奋的技术，包括将我们的图形转换为超图，以确保我们在原始图形中执行的某些过程在某种意义上“有效地”运行。总之，这些技术使我们能够证明新的结果，即几乎所有这些特殊颜色的图实际上都包含一条使用图的所有顶点的彩虹路径。我们还试图将这些想法应用到一个问题上，这个问题涉及到其他特殊有色图形中略有不同的彩虹结构。

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.