Project Summary Spanning cycles, perfect matchings, and other graph factors are fundamental concepts of graph theory that have been intensively studied. Seminal results of Dirac, Hall, and Tutte are central to the theory and have admitted many applications. At the same time, natural questions of a similar nature that arise in the domain of directed graphs and hypergraphs are not well understood. Only very recently there has been some progress on addressing the very fundamental problems on the existence of spanning cycles in oriented graphs or spanning cycles in hypergraphs and in contrast to undirected graphs, our understanding of these central problems in the realm of directed graphs and hypergraphs is far from satisfactory. PI proposes to investigate extremal problems on the existence of factors in directed graphs and hypergraphs. The main meta-problem which reverberates throughout the proposal is to find conditions for the minimum degree of a digraph H (or a hypergraph H) that guarantee the existence of a specific spanning digraph (hypergraph) of H. Proposed research problems fall into two main categories: 1. Problems on directed graphs. 2. Problems on 3-uniform hypergraphs. Some questions in the first group lead to problems on weighted graphs and the solutions to the weighted counterparts give solutions to the original problems. As a result, in addition to problems on 2-factors in directed graphs, PI proposes questions on 2-factors in weighted graphs and shows how the two groups of topics are related. In addition, PI intends to study problems on spanning cycles in hypergraphs. Two different notions of cycles in hypergraphs are considered and problems on the minimum degree conditions that guarantee the existence of factors consisting of these cycles are proposed. The problems in both groups can be studied using methods stemming from the regularity lemma of Szemeredi. PI has obtained preliminary results for some of the problems and intends to further advance his approach to tackle more involved questions described in the proposal. The research described in this proposal will impact graph theory and can lead to a better understanding of fundamental questions underling the theory of directed graphs and hypergraphs. A
|Effective start/end date||5/30/13 → 9/30/15|
- DOD: National Security Agency (NSA): $49,377.00
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