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

这一行动资助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市的莫纳什大学开展了一个数学研究项目。该项目的标题是“图论中的新概念”。主持科学家是大卫·伍德。该项目专注于图次要理论领域中的几个问题，特别强调哈密顿性。图是一组顶点和一组连接某些顶点对的边。图H称为图G的子图，如果H可以通过删除和收缩图G的边由图G构成，如果图包含恰好通过图的每个顶点一次的圈，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，则称图为k-连通的。该项目研究了不含k2，t子项的3连通图的哈密顿性(其中k2，t是具有大小为2和t的划分集的完全二部图)，并且还寻找了所有k2，5-子项自由图的特征，因为所有K2，4-子项自由图的特征已经准备好了。此外，还求出了一类子闭图的最小禁止子式，其中每个图的每个子式都是次哈密尔顿平面图。一个图是次哈密尔顿平面图，如果它是一个平面哈密尔顿图的支撑子图，如果一类图是次闭的，如果这类图中每个图的每一个子图也在这个类中。如果一个图可以在平面上画而不交叉边，则它是平面的。最后，该项目检验了Barnette和Goodey的一个猜想，该猜想涉及具有某些面大小限制的3-连通平面图的哈密顿性。EAPSI奖学金的更广泛影响包括为研究员提供美国以外的第一手研究经验；介绍各自所在地的科学、科学政策和科学基础设施；以及了解社会、文化和语言。这些活动符合NSF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为国际合作进行教育，从而确保拥有一支具有全球意识的美国科学队伍。此外，该研究员将有机会与她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业已存在的联系，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.