5 research outputs found
A degree sequence strengthening of the vertex degree threshold for a perfect matching in 3-uniform hypergraphs
The study of asymptotic minimum degree thresholds that force matchings and
tilings in hypergraphs is a lively area of research in combinatorics. A key
breakthrough in this area was a result of H\`{a}n, Person and Schacht who
proved that the asymptotic minimum vertex degree threshold for a perfect
matching in an -vertex -graph is
. In this paper we improve on this
result, giving a family of degree sequence results, all of which imply the
result of H\`{a}n, Person and Schacht, and additionally allow one third of the
vertices to have degree below this threshold.
Furthermore, we show that this result is, in some sense, tight.Comment: 21 page
Extremal problems in graphs
In the first part of this thesis we will consider degree sequence results for graphs. An important result of Komlós [39] yields the asymptotically exact minimum degree threshold that ensures a graph contains an -tiling covering an -proportion of the vertices of (for any fixed (0, 1) and graph ). In Chapter 2, we give a degree sequence strengthening of this result. A fundamental result of Kühn and Osthus [46] determines up to an additive constant the minimum degree threshold that forces a graph to contain a perfect -tiling. In Chapter 3, we prove a degree sequence version of this result.
We close this thesis in the study of asymmetric Ramsey properties in . Specifically, for fixed graphs we study the asymptotic threshold function for the property → . Rödl and Ruciński [61, 62, 63] determined the threshold function for the general symmetric case; that is, when . Kohayakawa and Kreuter [33] conjectured the threshold function for the asymmetric case. Building on work of Marciniszyn, Skokan, Spöhel and Steger [51], in Chapter 4, we reduce the 0-statement of Kohayakawa and Kreuter’s conjecture to a more approachable, deterministic conjecture. To demonstrate the potential of this approach, we show our conjecture holds for almost all pairs of regular graphs (satisfying certain balancedness conditions)