Minors in large and highly connected graphs

理解图不变量之间的相互作用以及它们强迫局部子结构的能力是图论的一个基本问题。我们想要调查不变量色数、平均度和连通度如何能够强迫未成年人，特别是密集未成年人，以及它们在这方面如何彼此不同。尽管这些不变量的多样性，但关于它们强迫人口稠密的未成年人的能力，人们几乎没有定性的区别。然而，哈德威格猜想--图论中的中心开放问题之一--暗示了这种差异。最近人们注意到，由于Robertson和Seymour的结果，Kr-minorless图的结构特征似乎暗示，对于大的图，高连通度比高平均度具有更密集的次要条件。这些研究以前所未有的方式将图子论和连通论的方法结合在一起。在一个由这两个领域的代表组成的小组中，我们希望进一步研究这些事态发展，从而有助于更多地阐明哈德维格猜想的领域。

It is a fundamental concern of graph theory to understand the interaction of graph invariants and their ability to force local substructures. We want to investigate how the invariants chromatic number, average degree, and connectivity are able to force minors, in particular dense minors, and how they differ from each other in this respect. In spite of the diversity of these invariants, few qualitative differences are known concerning their power to force dense minors. However, Hadwiger’s conjecture, one of the central open questions in graph theory, implies that there is such a difference. It has recently been noticed that the structural characterizations of Kr-minorfree graphs due to Robertson and Seymour appear to imply that, for large graphs, high connectivity forces denser minors than high average degree does. These investigations combine methods of graph minor theory and connectivity theory in a new and unprecedented way. In a group consisting of representatives of both these fields we would like to look further into these developments and thus help to shed some more light on the area of Hadwiger’s conjecture.