Extremal problems in graphs

Abstract

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 $G$ contains an $H$-tiling covering an $x$-proportion of the vertices of $G$ (for any fixed $x ∈$ (0, 1) and graph $H$). 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 $H$-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 $G_n,_p$. Specifically, for fixed graphs $H_1, . . . , H_r,$ we study the asymptotic threshold function for the property $G_n,_p$ → $H_1, . . . , H_r$. Rödl and Ruciński [61, 62, 63] determined the threshold function for the general symmetric case; that is, when $H_1 = · · · = H_r$. 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)