Combinatorial games

我最成功的研究到目前为止一直是我的研究，连同丽贝卡赫尔曼，“扩散游戏”的图形，关于这一点，我们已经写了文件“一致的边界非负的扩散游戏”。扩散博弈是一个过程，在这个过程中，图的每个顶点都被分配了一个数字标签，代表了一些筹码。这些标签根据以下规则以离散整数时间步长更新：对于端点具有不同数量的芯片的每条边，我们（同时）将一个芯片从具有更多芯片的顶点转移到具有更少芯片的顶点。这一过程之前由Duffy，Lidbetter，Messinger和Nowakowski介绍，并由Long和Narayanan进一步研究。在去年发表在arXiv上的一篇论文中，Long和Narayanan提出了一个问题，即对于固定数量的顶点，每个顶点上的初始芯片数量是否会在任何后续时间步上给出顶点上芯片数量的恒定界限。我们完整地回答了这个问题，如果我们在每个顶点上至少有（n - 2）个芯片，那么每个顶点上的芯片数将保持非负。此外，这个界限是紧的。我们还建立了一个相关的（部分）界的基础上的最大程度的图，我们希望在这个问题上取得进一步的进展，在未来。这些结果和随后的问题，在我们前面提到的论文中详细讨论。

My most successful research so far has been my study, together with Rebekah Herrman, of the `Diffusion Game' on graphs, about which we have written the paper `Uniform Bounds for Non-negativity ofthe Diffusion Game'. The diffusion game is a process in which each vertex of a graph is assigned a numerical label, representing a number of chips. These labels are updated at discrete integer time steps according to the following rule: for every edge whose endpoints have differing numbers of chips, we (simultaneously) transfer one chip from the vertex with more chips to the vertex with fewer chips. This process had previously been introduced by Duffy, Lidbetter, Messinger and Nowakowski, and further studied by Long and Narayanan. In a paper posted to arXiv last year, Long and Narayanan posed the question of whether, for a fixed number of vertics, bounding the initial number of chips on each vertex would give a constant bound on the number of chips on a vertex at any subsequent time step. We answered this question completely, if we start with at least (n - 2) chips on each vertex, then the number of chips on each vertex will remain non- negative. Furthermore, this bound is tight. We also establish a related (partial) bound based on the maximum degree of the graph; we hope to make further progress on this question in the future. These results, and subsequent questions, are discussed in detail in our aforementioned paper.