Cycles of a given length in tournaments

We study the asymptotic behavior of the maximum number of directed cycles of a given length in a tournament: let $c(\ell)$ be the limit of the ratio of the maximum number of cycles of length $\ell$ in an $n$-vertex tournament and the expected number of cycles of length $\ell$ in the random $n$-vertex tournament, when $n$ tends to infinity. It is well-known that $c(3)=1$ and $c(4)=4/3$. We show that $c(\ell)=1$ if and only if $\ell$ is not divisible by four, which settles a conjecture of Day. If $\ell$ is divisible by four, we show that $1+2\cdot\left(2/\pi\right)^{\ell}\le c(\ell)\le 1+\left(2/\pi+o(1)\right)^{\ell}$ and determine the value $c(\ell)$ exactly for $\ell = 8$. We also give a full description of the asymptotic structure of tournaments with the maximum number of cycles of length $\ell$ when $\ell$ is not divisible by four or $\ell\in\{4,8\}$.Comment: One of the programs used to verify the validity of Lemma 16 is available as an ancillary fil

The Shifted Turan Sieve Method on Tournaments

This article has been published in a revised form in the Canadian Mathematical Bulletin http://dx.doi.org/10.4153/S000843951900016X. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. Copyright © Canadian Mathematical Society 2019.Abstract. We construct a shi ed version of the Turán sieve method developed by R. Murty and the second author and apply it to counting problems on tournaments. More precisely, we obtain upper bounds for the number of tournaments which contain a fixed number of restricted r-cycles. These are the first concrete results which count the number of cycles over “all tournaments”.Research partially supported by NSERC Discovery Grants || CAPES and CSF/CNPQ, Brazil

BOUNDING THE NUMBER OF COMPATIBLE SIMPLICES IN HIGHER DIMENSIONAL TOURNAMENTS

A tournament graph G is a vertex set V of size n, together with a directed edge set E ⊂ V × V such that (i, j) ∈ E if and only if (j, i) ∉ E for all distinct i, j ∈ V and (i, i) ∉ E for all i ∈ V. We explore the following generalization: For a fixed k we orient every k-subset of V by assigning it an orientation. That is, every facet of the (k − 1)-skeleton of the (n − 1)-dimensional simplex on V is given an orientation. In this dissertation we bound the number of compatible k-simplices, that is we bound the number of k-simplices such that its (k − 1)-faces with the already-specified orientation form an oriented boundary. We prove lower and upper bounds for all k ≥ 3. For k = 3 these bounds agree when the number of vertices n is q or q + 1 where q is a prime power congruent to 3 modulo 4. We also prove some lower bounds for values k \u3e 3 and analyze the asymptotic behavior

A Collection of Problems in Extremal Combinatorics

PhDExtremal combinatorics is concerned with how large or small a combinatorial structure can be if we insist it satis es certain properties. In this thesis we investigate four different problems in extremal combinatorics, each with its own unique flavour. We begin by examining a graph saturation problem. We say a graph G is H-saturated if G contains no copy of H as a subgraph, but the addition of any new edge to G creates a copy of H. We look at how few edges a Kp- saturated graph can have when we place certain conditions on its minimum degree. We look at a problem in Ramsey Theory. The k-colour Ramsey number Rk(H) of a graph H is de ned as the least integer n such that every k- colouring of Kn contains a monochromatic copy of H. For an integer r > 3 let Cr denote the cycle on r vertices. By studying a problem related to colourings without short odd cycles, we prove new lower bounds for Rk(Cr) when r is odd. Bootstrap percolation is a process in graphs that can be used to model how infection spreads through a community. We say a set of vertices in a graph percolates if, when this set of vertices start off as infected, the whole graph ends up infected. We study minimal percolating sets, that is, percolating sets with no proper percolating subsets. In particular, we investigate if there is any relation between the smallest and the largest minimal percolating sets in bounded degree graph sequences. A tournament is a complete graph where every edge has been given an orientation. We look at the maximum number of directed k-cycles a tournament can have and investigate when there exist tournaments with many more k-cycles than expected in a random tournament.Engineering and Physical Sciences Research Council, grant number EP/K50290X/1

Topics in Extremal and Probabilistic Combinatorics

In this thesis, we consider a collection of problems in extremal and probabilistic combinatorics, specifically graph theory. First, we consider a close relative of the Cops and Robbers game called Revolutionaries and Spies, a two-player pursuit/evasion game devised by Beck to model network security. We show that on a ‘typical’ graph, if the second player has fewer pieces than are required to execute a particular trivial winning strategy, then the game is a first player win. Second, we consider the emergence of the square of a Hamilton cycle in a random geometric graph process, and show that typically, the exact instant at which a simple local obstacle is eliminated at every vertex, is the exact instant at which the graph becomes square Hamiltonian. This is in stark contrast to the ‘normal’ Erdos-Rényi random graph process, in which square Hamiltonicity is both not ‘local' in this sense, and occurs only once the graph is reasonably dense. Finally, we study an extremal problem concerning tournaments, that of maximising the number of oriented cycles of a fixed length. A ‘folklore’ result states that for 3-cycles one cannot do significantly better than a random tournament. More recent work shows that same is true for 5-cycles, and perhaps surprisingly that this is not true for 4-cycles. We conjecture that one can significantly beat the random tournament in expectation if and only if the length of the cycle is divisible by four, proving the ‘if’ statement, as well as a variety of new cases of the ‘only if' statement, including the case that the graph is sufficiently close to being regular.Engineering and Physical Sciences Research Council grant EP/M506394/