Perfect Matchings, Tilings and Hamilton Cycles in Hypergraphs

This thesis contains problems in finding spanning subgraphs in graphs, such as, perfect matchings, tilings and Hamilton cycles. First, we consider the tiling problems in graphs, which are natural generalizations of the matching problems. We give new proofs of the multipartite Hajnal-Szemeredi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs. In particular, we determine the minimum codegree thresholds for Hamilton l-cycles in large k-uniform hypergraphs for l less than k/2. We also determine the minimum vertex degree threshold for loose Hamilton cycle in large 3-uniform hypergraphs. These results generalize the well-known theorem of Dirac for graphs. Third, we determine the minimum codegree threshold for near perfect matchings in large k-uniform hypergraphs, thereby confirming a conjecture of Rodl, Rucinski and Szemeredi. We also show that the decision problem on whether a k-uniform hypergraph with certain minimum codegree condition contains a perfect matching can be solved in polynomial time, which solves a problem of Karpinski, Rucinski and Szymanska completely. At last, we determine the minimum vertex degree threshold for perfect tilings of C_4^3 in large 3-uniform hypergraphs, where C_4^3 is the unique 3-uniform hypergraph on four vertices with two edges

Hamilton cycles in graphs and hypergraphs: an extremal perspective

As one of the most fundamental and well-known NP-complete problems, the Hamilton cycle problem has been the subject of intensive research. Recent developments in the area have highlighted the crucial role played by the notions of expansion and quasi-randomness. These concepts and other recent techniques have led to the solution of several long-standing problems in the area. New aspects have also emerged, such as resilience, robustness and the study of Hamilton cycles in hypergraphs. We survey these developments and highlight open problems, with an emphasis on extremal and probabilistic approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page limits, this final version is slightly shorter than the previous arxiv versio

The minimum vertex degree for an almost-spanning tight cycle in a 33-uniform hypergraph

We prove that any $3$-uniform hypergraph whose minimum vertex degree is at least $\left(\frac{5}{9} + o(1) \right)\binom{n}{2}$ admits an almost-spanning tight cycle, that is, a tight cycle leaving $o(n)$ vertices uncovered. The bound on the vertex degree is asymptotically best possible. Our proof uses the hypergraph regularity method, and in particular a recent version of the hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1411.495

Loose Hamiltonian cycles forced by large (k−2)(k-2)-degree - sharp version

We prove for all $k\geq 4$ and $1\leq\ell<k/2$ the sharp minimum $(k-2)$-degree bound for a $k$-uniform hypergraph $\mathcal H$ on $n$ vertices to contain a Hamiltonian $\ell$-cycle if $k-\ell$ divides $n$ and $n$ is sufficiently large. This extends a result of Han and Zhao for $3$-uniform hypegraphs.Comment: 14 pages, second version addresses changes arising from the referee report

Hamilton cycles in quasirandom hypergraphs

We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $\Omega(n^{k-1})$ contains a loose Hamilton cycle. We also give a construction to show that a $k$-uniform hypergraph satisfying these conditions need not contain a Hamilton $\ell$-cycle if $k-\ell$ divides $k$. The remaining values of $\ell$ form an interesting open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm