NSF East Asia and Pacific Summer Institute (EAPSI) for FY 2013 in Australia

这项行动资助了范德比尔特大学的艾米丽·马歇尔于2013年夏天在澳大利亚维多利亚州克莱顿的莫纳什大学进行一项数学研究项目。这个项目的题目是“图论中的新概念”。主持会议的科学家是大卫·伍德。该项目侧重于图小理论领域的几个问题，特别强调哈密顿性。图是一组顶点和一组连接一些顶点对的边的集合。如果H可以由G通过删除和收缩G的边而形成，那么图H就被称为图G的次元。如果图H包含一个经过图中每个顶点一次的循环，则称为哈密顿图。如果图中任意k-1个顶点的移除不会断开图的连接，则称为k连通图。该项目检查了关于哈密性的没有K2，t次的3连通图（其中K2，t是具有大小为2和t的分割集的完全二部图），并且还寻找所有K2,5次自由图的表征，因为所有K2,4次自由图的表征已经在准备中。此外，它还试图确定一类特定的小闭图的最小禁止子图，其中每个图的每个子图都是次哈密顿平面。如果一个图是一个平面哈密顿图的生成子图，那么这个图就是次哈密顿平面图；如果一类图中每个图的每个次图也在该类中，那么这类图就是次闭图。如果一个图形可以在平面上画而没有交叉边，那么这个图形就是平面的。最后，研究了Barnette和Goodey提出的一个猜想，该猜想涉及具有一定面尺寸限制的3连通平面图的哈密顿性。EAPSI奖学金的广泛影响包括为研究员提供美国以外的第一手研究经验；介绍该地区的科学、科学政策和科学基础设施；以及对社会、文化和语言的了解。这些活动符合美国国家科学基金会的目标，即在其科学家、工程师和教育工作者的职业生涯早期培养国际合作，从而确保美国科学劳动力具有全球意识。此外，该研究员将有机会与其所在机构的成员分享该项目的成果。该项目将加强范德比尔特大学和墨尔本大学之间已经存在的联系，为未来的国际合作铺平道路。

This action funds Emily Marshall of Vanderbilt University to conduct a research project in Mathematics during the summer of 2013 at Monash University in Clayton, Victoria, Australia. The project title is "Novel Concepts in Graph Theory." The host scientist is David Wood. The project focuses on several problems in the area of graph minor theory with a particular emphasis on Hamiltonicity. A graph is a set of vertices together with a set of edges which connect some pairs of vertices. A graph H is said to be a minor of a graph G if H can be formed from G by deleting and contracting edges of G. A graph is called Hamiltonian if it contains a cycle which passes through every vertex of the graph exactly once. A graph is called k-connected if the removal of any set of k-1 vertices in the graph never disconnects the graph. The project examines 3-connected graphs with no K2,t minors in regards to Hamiltonicity (where K2,t is the complete bipartite graph with partition sets of size 2 and t), and also looks for a characterization of all K2,5-minor free graphs since a characterization of all K2,4-minor free graphs is already in preparation. In addition, it seeks to determine the minimal forbidden minors of a specific class of minor-closed graphs in which every minor of every graph is subhamiltonian planar. A graph is subhamiltonian planar if it is the spanning subgraph of a planar Hamiltonian graph, and a class of graphs is minor-closed if every minor of every graph in the class is also in the class. A graph is planar if it can be drawn on the plane without crossing edges. Finally, the project examines a conjecture due to Barnette and Goodey involving the Hamiltonicity of 3-connected planar graphs with certain face size restrictions. Broader impacts of an EAPSI fellowship include providing the Fellow a first-hand research experience outside the U.S.; an introduction to the science, science policy, and scientific infrastructure of the respective location; and an orientation to the society, culture and language. These activities meet the NSF goal to educate for international collaborations early in the career of its scientists, engineers, and educators, thus ensuring a globally aware U.S. scientific workforce. Furthermore, the Fellow will have the opportunity to share the results of the project with members of her home institution. The project will strengthen already existing ties between Vanderbilt University and universities in Melbourne paving the way for future international collaboration.