Georgia Institute of Technology College of Sciences

Kühn, Osthus, and Treglown and, independently, Khan proved that if H is a 3-uniform hypergraph on n vertices, where n is a multiple of 3 and large, and the minimum vertex degree of H is greater than {(n-1) choose 2} - {2n/3 choose 2}, then H contains a perfect matching.

We show that for sufficiently large n divisible by 3, if F_1, ..., F_{n/3} are 3-uniform hypergraphs with a common vertex set and the minimum vertex degree in each F_i is greater than {(n-1) choose 2} - {2n/3 choose 2} for i = 1, ..., n/3, then the family {F_1, ..., F_{n/3}} admits a rainbow matching, i.e., a matching consisting of one edge from each F_i. This is done by converting the rainbow matching problem to a perfect matching problem in a special class of uniform hypergraphs.

We also prove that, for any integers k, l with k >= 3 and k/2 < l <= k-1, there exists a positive real μ such that, for all sufficiently large integers m, n satisfying n/k - μn <= m <= n/k - 1 - (1 - l/k){ceil of (k - l)/(2l - k)}, if H is a k-uniform hypergraph on n vertices and the minimum l-degree of H is greater than {(n-l) choose (k-l)} - {(n-l-m) choose (k-l)}, then H has a matching of size m+1. This improves upon an earlier result of Hàn, Person, and Schacht for the range k/2 < l <= k-1.  In many cases, our result gives tight bound on the minimum l-degree of H for near perfect matchings. For example, when l >= 2k/3, n ≡ r (mod k), 0 <= r < k, and r + l >= k, we can take m to be the minimum integer at least n/k - 2.

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Erdos and Posa proved in 1965 that cycles satisfy an approximate packing-covering duality. Finding analogous approximate dualities for other families of graphs has since become a highly active area of research due in part to its algorithmic applications. In this thesis we investigate the Erdos-Posa property of various families of constrained cycles and paths by developing new structural tools for undirected group-labelled graphs.

Our first result is a refinement of the flat wall theorem of Robertson and Seymour to undirected group-labelled graphs. This structure theorem is then used to prove the Erdos-Posa property of A-paths of length 0 modulo p for a fixed odd prime p, answering a question of Bruhn and Ulmer. Further, we obtain a characterization of the abelian groups G and elements g for which A-paths of weight g satisfy the Erdos-Posa property. These results are from joint work with Robin Thomas.

We extend our structural tools to graphs labelled by multiple abelian groups and consider the Erdos-Posa property of cycles whose weights avoid a fixed finite subset in each group. We find three types of topological obstructions and show that they are the only obstructions to the Erdos-Posa property of such cycles. This is a far-reaching generalization of a theorem of Reed that Escher walls are the only obstructions to the Erdos-Posa property of odd cycles. Consequently, we obtain a characterization of the sets of allowable weights in this setting for which the Erdos-Posa property holds for such cycles, unifying a large number of results in this area into a general framework. As a special case, we characterize the integer pairs (L,M) for which cycles of length L mod M satisfy the Erdos-Posa property. This resolves a question of Dejter and Neumann-Lara from 1987. Further, our description of the obstructions allows us to obtain an analogous characterization of the Erdos-Posa property of cycles in graphs embeddable on a fixed compact orientable surface. This is joint work with Pascal Gollin, Kevin Hendrey, O-joung Kwon, and Sang-il Oum.

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