An approximate Dirac-type theorem for k -uniform hypergraphs

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-Szemerédi Theorem for the tripartite and quadripartite cases. Second, we consider Hamilton cycles in hypergraphs....

Recent Advances on Dirac-type Problems for Hypergraphs

A fundamental question in graph theory is to establish conditions that ensure a graph contains certain spanning subgraphs. Two well-known examples are Tutte’s theorem on perfect matchings and Dirac’s theorem on Hamilton cycles. Generalizations of Dirac’s theorem, and related matching and packing problems for hypergraphs, have received much attention in recent years. New tools such as the absorb...

A Dirac-Type Theorem for 3-Uniform Hypergraphs

A Dirac-type theorem for Hamilton Berge cycles in random hypergraphs

A Hamilton Berge cycle of a hypergraph on n vertices is an alternating sequence (v1, e1, v2, . . . , vn, en) of distinct vertices v1, . . . , vn and distinct hyperedges e1, . . . , en such that {v1, vn} ⊆ en and {vi, vi+1} ⊆ ei for every i ∈ [n − 1]. We prove a Dirac-type theorem for Hamilton Berge cycles in random r-uniform hypergraphs by showing that for every integer r ≥ 3 there exists k = k...

The Complexity of the Hamilton Cycle Problem in Hypergraphs of High Minimum Codegree

We consider the complexity of the Hamilton cycle decision problem when restricted to k-uniform hypergraphs H of high minimum codegree δ(H). We show that for tight Hamilton cycles this problem is NP-hard even when restricted to k-uniform hypergraphsH with δ(H) ≥ n2−C, where n is the order of H and C is a constant which depends only on k. This answers a question raised by Karpiński, Ruciński and ...