This section gives a short outline of the structure of the remainder of the thesis as well as an introduc-tion to the notaintroduc-tion and syntax used therein. In Chapter 2 we introduce the main concepts, explain related work, and then present, discuss, and analyse our results in detail. Afterwards, in Chapters 3–
7 we give the proofs of our theorems. Concluding remarks and a brief discussion of open problems complete the thesis in Chapter 8.
We want to remark at this point that this thesis is a cumulation of the results from four different papers [4, 32, 64, 94], each with a different set of coauthors. The extension of Riordan’s theorem from [94] also appeared previously as an extended abstract [93], the main result from [32] appeared in [31], and the result from [6] in [5]. In all four papers, the author of this thesis contributed
signifi-1.3 Preliminaries and notation
cantly to all stages, beginning from the research conducted, until the preparation of the final paper.
Large parts of this thesis will be verbatim copies from the papers, in particular Chapter 3 from [94], Chapter 4 from [4], Chapter 5 from [32], Chapter 6 from [94] and Chapter 7 from [64] with a short paragraph from [94]. Moreover, the following notation, the abstract, German summary, some para-graphs explaining the respective results in Chapter 2 and short parts of the Conclusion in Chapter 8 are close adaptations from the corresponding parts in [4, 32, 64, 94]. All these results do not appear in any other thesis.
To state the results in detail, avoid too much repetition, and have a point of reference for the reader we now collect the basic notation. We mostly follow the standard notation from [30, 60, 65] and thus the reader who is familiar with the basics can skip the rest of this section. We state the definitions for hypergraphs and remark that withr = 2 this gives the corresponding definitions for graphs, in which case we usually omit the superscript. Anr-uniform hypergraphH is a tuple(V, E), where V(H) :=V is its vertex set andE(H) :=E ⊆ Vr
is the set of edges inH. We writev(H)for|V(H)|
ande(H)for|E(H)|. ByKn(r)we denote thecompleter-uniform hypergraph [n], nr
on the vertex set [n] :={1,2, . . . , n}. Therandomr-uniform hypergraphH(r)(n, p)is the probability space of all labelled r-uniform hypergraphs on the vertex set[n], where each edgee∈ [n]r
is chosen independently of all the other edges with probabilityp.
We say that a graphH contains a graphGas asubgraphif there exists a mapφfromV(G)toV(H) such that edges are preserved, i.e. for alle∈E(G)we have thatφ(e)∈E(H). IfGis a subgraph ofH we writeG⊆H. The subgraphinducedby a subset of the verticesW ⊆V inHis denoted byH[W] :=
W, E(H)∩ Wr
and we defineH−W =H[V\W]. We denote bydegH(f) :=|{e:f ⊆e}|thedegree of a set of verticesfof size1≤ |f| ≤r−1inH, i.e. the number of edgesfis contained in. Given a set W ⊆V, we writedegH(f, W)for the degree intoW, that is, we count only edgesesatisfyinge\f ⊂W. Further,∆`(H)is defined to be themaximum`-degreeinH, i.e.∆`(H) := max{degH(f) :f ∈ V`
}.
We usually omit the subscript if ` = 1. With d(H) := v(H)e(H) and d1(H) := v(H)−1e(H) we define the densitym(H) := max{d(H0) :H0 ⊆H} and the one-densitym1(H) := max{d1(H0) :H0⊆H}. The shadow graphH0 is obtained fromH by replacing every edgee∈E(H)by all possible r2
subsets of cardinality two (ignoring multiple edges).
An alternating sequence of vertices and edgesv1, e1, v2, e2, . . . , vt, et, vt+1is called apath7of length tfromv1tovt+1ifvi,vi+1∈eifor alli∈[t]. If there is a path fromutov, then we say thatuandvare connected. This defines an equivalence relation onV. We say that a hypergraphH isconnectedif there is a path between any two vertices ofH. Acomponentin anr-uniform hypergraph is a maximally connected subgraph. Thedistancebetween two verticesuandvinH is the minimal length over all paths fromutov, and if they are in different components then we set it to infinity.
TheneighbourhoodNH(v)of a vertexvis the set of vertices which are contained in an edge together withv
NH(v) :={w∈V \ {v}:∃e∈Es.t.{w, v} ⊆e}.
For a subset of the verticesW ⊆V, the neighbourhood inHisNH(W) =S
w∈W NH(w)\W. If there is
7In our definition of a path, we do not mind repetitions of vertices and edges. Usually, this is referred to as awalkrather than a path.
4
1. Introduction
no risk of confusion, we sometimes omit the graph in the subscript. The setW is calledt-independent in a hypergraphH, if the distance betweenv∈W andw∈W inH is at leastt+ 1. A1-independent set is independent in the usual sense.
Letf andg be real valued functions, where we usually omit the dependencies on variables. We writeg =O(f)8ifg is not growing much faster thanf, i.e. there existC > 0andn0 >0such that for alln > n0 we have|f(n)| ≤ C· |g(n)|. Iff = O(g), then vice versag is not decreasing much faster thanf and thus we can also writeg= Ω(f), and if both statements holdg= Θ(f). We usually make no effort in optimising the constantC hidden in this notation. Sometimes we include other constants as a subscript to point out that C depends on them. Byω(f)we denote any functiong that is growing faster thanf, i.e.g(n)/f(n)→ ∞asn→ ∞, and similarlyo(g)denotes any function f such thatg = ω(f). For brevity we will often useω(1)instead of saying that there exists a large enough constantC, even though this is is general.
Withlnnwe denote the natural logarithm, bute we usually uselognif the base does not matter.
With a polylog-factor orpolylog nwe refer to any polynomial inlogn. To simplify readability, we will omit in the calculations floor and ceiling signs whenever they are not crucial for the arguments.
8Formally it should beg∈O(f)but it is common pratice to abuse the notation in this way.
Chapter 2
Results, discussion and outline
This is the main chapter of this thesis as all our results are motivated and explained. We will also give brief sketches of the proofs and include a short discussion on the methods and techniques that are involved, but the rigorous proofs will be found in later chapters of the thesis. In order to embed the theorems into a wider context, we will introduce some established concepts and well-known results which will eventually lead to our main results. We will guide the reader from single spanning structures, via algorithmic results for Hamilton cycles and the randomly perturbed graphs model, to universality. The last part of this chapter is used to introduce standard tools that we will make extensive use of in the remainder of the thesis.
2.1 Thresholds
A graph propertyFis a set of graphs. This set could for example consist of all graphs with a specific subgraph, special structure, fixed chromatic number or any other graph parameter. Consider, for example, the graph propertyFHAM of having aHamilton cycle as a subgraph, then a graphH is in FHAMif there exists a cyclic ordering of the vertices ofHsuch that neighbouring vertices are adjacent.
It is a classical result of Dirac [42] that every graph onnvertices with minimum degree at leastn/2 satisfies this property and thus is Hamiltonian. This result is sharp in the sense that there are graphs with minimum degree slightly belown/2that do not contain a Hamilton cycle and thus minimum degreen/2is a distinguished point.
In random graph theory one of the most natural objectives is to study similar graph properties in G(n, p)as previously in deterministic graphs. For example we can ask when doesG(n, p)contain a Hamilton cycle. More precisely, we can ask for which values ofpdoesP[G(n, p) ∈ F]tend to1 as ntends to infinity for some propertyF. If this is true, then we say thatG(n, p)has the propertyF asymptotically almost surely(a.a.s.).
Containing a subgraph is amonotone property, which means that adding edges cannot destroy the property and thus with larger pthe probability that G(n, p) has this property increases. With this observation in mind it makes sense to ask for a valuepˆbelow whichG(n, p)does not have the property a.a.s. but above it has. It turns out that often there is a very abrupt change in behaviour and thus we say thatpˆ:N→[0,1]is athreshold functionfor a graph propertyFif
P
G(n, p)∈ F
→0 ifp=o(ˆp)
→1 ifp=ω(ˆp).
Sometimes this kind of threshold is referred to ascoarse, where for asharpthreshold we require for