Large structures in dense directed graphs

We answer questions in extremal combinatorics, for directed graphs. Specifically, we investigate which large tree-like directed graphs are contained in all dense directed graphs of large order. More precisely, let T be an oriented tree of order n; among others, we establish the following results. (1) We obtain a sufficient condition which ensures every tournament of order n contains T, and show that almost every tree possesses this property. (2) We prove that for all positive C, ɛ and sufficiently large n, every tournament of order (1+ɛ)n contains T if Δ(T)≤(log n)^C. (3) We prove that for all positive Δ, ɛ and sufficiently large n, every directed graph G of order n and minimum semidegree (1/2+ɛ)n contains T if Δ(T)≤Δ. (4) We obtain a sufficient condition which ensures that every directed graph G of order n with minimum semidegree at least (1/2+ɛ)n contains T, and show that almost every tree possesses this property. (5) We extend our method in (4) to a class of tree-like spanning graphs which includes all orientations of Hamilton cycles and large subdivisions of any graph. Result (1) confirms a conjecture of Bender and Wormald and settles a conjecture of Havet and Thomassé for almost every tree; (2) strengthens a result of Kühn, Mycroft and Osthus; (3) is a directed graph analogue of a classical result of Komlós, Sárközy and Szemerédi and is implied by (4) and (5) is of independent interest

Diamond-free partial orders

This thesis presents initial work in attempting to understand the class of ‘diamond-free’ 3-cs-transitive partial orders. The notion of diamond-freeness, proposed by Gray, says that for any a ≤ b, the set of points between a and b is linearly ordered. A weak transitivity condition called ‘3-cs-transitivity’ is taken from the corresponding notion for cycle-free partial orders, which in that case led to a complete classification [3] of the countable examples. This says that the automorphism group acts transitively on certain isomorphism classes of connected 3-element structures. Classification for diamond-free partial orders seems at present too ambitious, but the strategy is to seek classifications of natural subclasses, and to test conjectures suggested by motivating examples. The body of the thesis is divided into three main inter-related chapters. The first of these, Chapter 3, adopts a topological approach, focussing on an analogue of topological covering maps. It is noted that the class of ‘covering projections’ between diamond-free partial orders can add symmetry or add cycles, and notions such as path connectedness transfer directly. The concept of the ‘nerve’ of a partial order makes this analogy concrete, and leads to useful observations about the fundamental group and the existence of an underlying cycle-free partial order called the universal cover. In Chapter 4, the work of [1] is generalised to show how to decompose ranked diamond- free partial orders. As in the previous chapter, any diamond-free partial order is covered by a specific cycle-free partial order. The paper [1] constructs a diamond-free partial order with cycles of height 1 from a different cycle-free partial order through which the universal covering factors. This is extended to construct a sequence of diamond- free partial orders with cycles of finite height which are not only factors but have the chosen diamond-free partial order as a ‘limit’. This leads to a better understanding of why structures with cycles only of height 1 are special, and the rest divide into structures with cycles of bounded height and a cycle-free backbone, and those for which the cycles have cofinal height. Even these can be expressed as limits of structures with cycles of 6 bounded height, though not directly. A variety of constructions are presented in Chapter 5, based on an underlying cycle- free partial order, and an ‘anomaly’, which in the simplest case given in [5] is a 2-level Dedekind-MacNeille complete 3-cs-transitive partial order, but which here is allowed to be a partial order of greater complexity. A rich class of examples is found, which have very high degrees of homogeneity and help to answer a number of conjectures in the negative

Transversal Problems In Sparse Graphs

Graph transversals are a classical branch of graph algorithms. In such a problem, one seeks a minimum-weight subset of nodes in a node-weighted graph $G$ which intersects all copies of subgraphs~$F$ from a fixed family $\mathcal F$. In the first portion of this thesis we show two results related to even cycle transversal. %%Note rephrase this later. In Chapter \ref{ECTChapter}, we present our 47/7-approximation for even cycle transversal. To do this, we first apply a graph ``compression" method of Fiorini et al. % \cite{FioriniJP2010} which we describe in Chapter \ref{PreliminariesChapter}. For the analysis we repurpose the theory behind the 18/7-approximation for ``uncrossable" feedback vertex set problems of Berman and Yaroslavtsev %% \cite{BermanY2012} noting that we do not need the larger ``witness" cycles to be a cycle that we need to hit. This we do in Chapter \ref{BermanYaroChapter}. In Chapter \ref{ErdosPosaChapter} we present a simple proof of an Erdos Posa result, that for any natural number $k$ a planar graph $G$ either contains $k$ vertex disjoint even cycles, or a set $X$ of at most $9k$ such that $G \backslash X$ contains no even cycle. In the rest of this thesis, we show a result for dominating set. A dominating set $S$ in a graph is a set of vertices such that each node is in $S$ or adjacent to $S$. In Chapter 6 we present a primal-dual $(a+1)$-approximation for minimum weight dominating set in graphs of arboricity $a$. At the end, we propose five open problems for future research

Interaction Topologies and Information Flow

Networks are ubiquitous, underlying systems as diverse as the Internet, food webs, societal interactions, the cell, and the brain. Of crucial importance is the coupling of network structure with system dynamics, and much recent attention has focused on how information, such as pathogens, mutations, or ideas, ow through networks. In this dissertation, we advance the understanding of how network structure a ects information ow in two important classes of models. The rst is an independent interaction model, which is used to investigate the propagation of advantageous alleles in evolutionary algorithms. The second is a threshold model, which is used to study the dissemination of ideas, fads, and innovations throughout populations. This journal-format dissertation comprises three interrelated studies, in which we investigate the in uence of network structure on the dynamical properties of information ow. In the rst study, we develop an analytical technique to approximate system dynamics in arbitrarily structured regular interaction topologies. In the second study, we investigate the ow of advantageous alleles in degree-correlated scale-free population structures, and provide a simple topological metric for assessing the selective pressures induced by these networks. In the third study, we characterize the conditions in which global information cascades occur in threshold models of binary decisions with externalities, structured on degree-correlated Poisson-distributed random networks

Symmetry in Graph Theory

This book contains the successful invited submissions to a Special Issue of Symmetry on the subject of ""Graph Theory"". Although symmetry has always played an important role in Graph Theory, in recent years, this role has increased significantly in several branches of this field, including but not limited to Gromov hyperbolic graphs, the metric dimension of graphs, domination theory, and topological indices. This Special Issue includes contributions addressing new results on these topics, both from a theoretical and an applied point of view