Decomposing graphs into edges and triangles

We prove the following 30 year-old conjecture of Győri and Tuza: the edges of every n-vertex graph G can be decomposed into complete graphs C1,. . .,Cℓ of orders two and three such that |C1|+···+|Cℓ| ≤ (1/2+o(1))n2. This result implies the asymptotic version of the old result of Erdős, Goodman and Pósa that asserts the existence of such a decomposition with ℓ ≤ n2/4

Sharp bounds for decomposing graphs into edges and triangles

Let pi3(G) be the minimum of twice the number of edges plus three times the number of triangles over all edge-decompositions of G into copies of K2 and K3. We are interested in the value of pi3(n), the maximum of pi3(G) over graphs G with n vertices. This specific extremal function was first studied by Gyori and Tuza [Decompositions of graphs into complete subgraphs of given order, Studia Sci. Math. Hungar. 22 (1987), 315--320], who showed that pi3(n)\u3c9n2/16.In a recent advance on this problem, Kral, Lidicky, Martins and Pehova [arXiv:1710:08486] proved via flag algebras that pi3(n)\u3c(1/2+o(1))n2, which is tight up to the o(1) term.We extend their proof by giving the exact value of pi3(n) for large n, and we show that Kn and Kn/2,n/2 are the only extremal examples

Compactness and finite forcibility of graphons

Graphons are analytic objects associated with convergent sequences of graphs. Problems from extremal combinatorics and theoretical computer science led to a study of graphons determined by finitely many subgraph densities, which are referred to as finitely forcible. Following the intuition that such graphons should have finitary structure, Lovasz and Szegedy conjectured that the topological space of typical vertices of a finitely forcible graphon is always compact. We disprove the conjecture by constructing a finitely forcible graphon such that the associated space is not compact. The construction method gives a general framework for constructing finitely forcible graphons with non-trivial properties

C5C_5 is almost a fractalizer

We determine the maximum number of induced copies of a 5-cycle in a graph on $n$ vertices for every $n$. Every extremal construction is a balanced iterated blow-up of the 5-cycle with the possible exception of the smallest level where for $n=8$, the M\"obius ladder achieves the same number of induced 5-cycles as the blow-up of a 5-cycle on 8 vertices. This result completes work of Balogh, Hu, Lidick\'y, and Pfender [Eur. J. Comb. 52 (2016)] who proved an asymptotic version of the result. Similarly to their result, we also use the flag algebra method but we extend its use to small graphs.Comment: 24 page

Maximizing five-cycles in Kr-free graphs

The Erdos Pentagon problem asks to find an n-vertex triangle-free graph that is maximizing the number of 5-cycles. The problem was solved using flag algebras by Grzesik and independently by Hatami, Hladky, Kral, Norin, and Razborov. Recently, Palmer suggested the general problem of maximizing the number of 5-cycles in K_{k+1}-free graphs. Using flag algebras, we show that every K_{k+1}-free graph of order n contains at most 110k4(k4−5k3+10k2−10k+4)n5+o(n5) copies of C_5 for any k≥3, with the Turan graph begin the extremal graph for large enough n

Theory of combinatorial limits and extremal combinatorics

In the past years, techniques from different areas of mathematics have been successfully applied in extremal combinatorics problems. Examples include applications of number theory, geometry and group theory in Ramsey theory and analytical methods to different problems in extremal combinatorics. By providing an analytic point of view of many discrete problems, the theory of combinatorial limits led to substantial results in many areas of mathematics and computer science, in particular in extremal combinatorics. In this thesis, we explore the connection between combinatorial limits and extremal combinatorics. In particular, we prove that extremal graph theory problemsmay have unique optimal solutions with arbitrarily complex structure, study a property closely related to Sidorenko's conjecture, one of the most important open problems in extremal combinatorics, and prove a 30-year old conjecture of Gyori and Tuza regarding decomposing the edges of a graph into triangles and edges

Packings and tilings in dense graphs

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