Packings and tilings in dense graphs

Abstract

In this thesis we present results on selected problems from extremal graph theory, and discuss both known and new methods used to solve them. In Chapter 1, we give an introductory overview of the regularity method, the flag algebra framework, and some probabilistic tools, which we use to prove our results in subsequent chapters. In Chapter 2 we prove a new result on the packing density of triangles in graphs with given edge density. In doing so, we settle a few conjectures of Gyori and Tuza on decompositions and coverings of graphs with cliques of bounded size. In Chapter 3 we show that a famous conjecture on Hamilton decompositions of bipartite tournaments due to Jackson holds approximately, providing the first intermediate result towards a full proof of the conjecture. In Chapter 4, we introduce a novel absorbing paradigm for graph tilings, which we apply in a few different settings to obtain new results. Using this method, we are able to extend a result on triangle-tilings in graphs with high minimum degree and sublinear independence number to clique-tilings of arbitrary size. We also strengthen an existing result on tilings in randomly perturbed graphs. Finally, in Chapter 5, we consider a problem on quasi-randomness in permutations. We obtain simple density conditions for a sequence of permutations to be quasirandom, and give a full characterisation of all conditions of the same type that force quasi-randomness in the same way