15 research outputs found
Completion and deficiency problems
Given a partial Steiner triple system (STS) of order , what is the order
of the smallest complete STS it can be embedded into? The study of this
question goes back more than 40 years. In this paper we answer it for
relatively sparse STSs, showing that given a partial STS of order with at
most triples, it can always be embedded into a complete
STS of order , which is asymptotically optimal. We also obtain
similar results for completions of Latin squares and other designs.
This suggests a new, natural class of questions, called deficiency problems.
Given a global spanning property and a graph , we define the
deficiency of the graph with respect to the property to be
the smallest positive integer such that the join has property
. To illustrate this concept we consider deficiency versions of
some well-studied properties, such as having a -decomposition,
Hamiltonicity, having a triangle-factor and having a perfect matching in
hypergraphs.
The main goal of this paper is to propose a systematic study of these
problems; thus several future research directions are also given
Cover 3-uniform hypergraphs by vertex-disjoint tight paths
Let be an -vertex 3-graph such that every pair of vertices is in at
least edges. We show that contains two vertex-disjoint tight
paths whose union covers the vertex set of . The quantity two here is best
possible and the degree condition is asymptotically best possible.Comment: 16 page
Graphs with few spanning substructures
In this thesis, we investigate a number of problems related to spanning substructures of graphs. The first few chapters consider extremal problems related to the number of forest-like structures of a graph. We prove that one can find a threshold graph which contains the minimum number of spanning pseudoforests, as well as rooted spanning forests, amongst all graphs on n vertices and e edges. This has left the open question of exactly which threshold graphs have the minimum number of these spanning substructures. We make progress towards this question in particular cases of spanning pseudoforests.
The final chapter takes on a different flavor---we determine the complexity of a problem related to Hamilton cycles in hypergraphs. Dirac\u27s theorem states that graphs with minimum degree at least half the size of the vertex set are guaranteed to have a Hamilton cycle. In 1993, Karpinksi, Dahlhaus, and Hajnal proved that for any c\u3c1/2, the problem of determining whether a graph with minimum degree at least cn has a Hamilton cycle is NP-complete. The analogous problem in hypergraphs, for both a Dirac-type condition and complexity, are just as interesting. We prove that for classes of hypergraphs with certain minimum vertex degree conditions, the problem of determining whether or not they contain an l-Hamilton cycle is NP-complete.
Advisor: Professor Jamie Radcliff
Dirac-type conditions for spanning bounded-degree hypertrees
We prove that for fixed , every -uniform hypergraph on vertices and
of minimum codegree at least contains every spanning tight -tree
of bounded vertex degree as a sub\-graph. This generalises a well-known result
of Koml\'os, S\'ark\"ozy and Szemer\'edi for graphs. Our result is
asymptotically sharp. We also prove an extension of our result to hypergraphs
that satisfy some weak quasirandomness conditions
Resilience for Loose Hamilton Cycles
We study the emergence of loose Hamilton cycles in subgraphs of random
hypergraphs. Our main result states that the minimum -degree threshold for
loose Hamiltonicity relative to the random -uniform hypergraph
coincides with its dense analogue whenever . The
value of is approximately tight for . This is particularly
interesting because the dense threshold itself is not known beyond the cases
when .Comment: 33 pages, 3 figure