Combinatorial games

到目前为止，我最成功的研究是我和丽贝卡·赫尔曼对图上的扩散博弈的研究，我们写了一篇论文《扩散博弈的非负性的统一界限》。扩散博弈是一个过程，在这个过程中，一个图的每个顶点都被分配了一个数字标签，代表许多筹码。这些标签按照以下规则以离散的整数时间步长进行更新：对于其端点具有不同码片数量的每条边，我们(同时)将一个码片从码片较多的顶点转移到码片较少的顶点。这一过程此前由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：对于固定数量的顶点，限制每个顶点上的初始筹码数量是否会在随后的任何时间步中给出一个顶点上筹码数量的恒定界限。我们完全回答了这个问题，如果我们在每个顶点上开始至少有(n-2)个筹码，那么每个顶点上的筹码数量将保持非负。此外，这一界限很紧。我们还建立了一个基于图的最大度的相关(部分)界，希望在以后的研究中取得进一步的进展。这些结果以及随后的问题将在前面的论文中详细讨论。

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.