The thesis is organised as follows.
In Chapter 2 we introduce the notation and definitions that we frequently use in the thesis. Moreover, we summarise necessary background material concerning the regularity method and analytic combinatorics, and state concentration inequalities for various probabil-ity distributions.
Next in Chapter 3 we prove analogues of the bandwidth theorem for sparse random and pseudorandom graphs, as well as a version for the embedding of degenerate graphs in sparse random graphs. These results are based on joint work with Peter Allen, Julia B¨ottcher, and Anusch Taraz [7, 8].
Then in Chapter 4 we present the proof of a Dirac-type theorem for Hamilton Berge cycles in randomr-uniform hypergraphs, investigate the problem for dense r-uniform hypergraphs, and prove bounds on the biases for which Avoider can keep his hypergraph (almost) Berge-acyclic in monotone and strict Avoider-Enforcer games played on the edge set of a complete 3-uniform hypergraph. The first part is joint work with Dennis Clemens and Yury Person [54]
and the second part extends a joint work with Dennis Clemens, Yury Person, and Tuan Tran [55].
Chapter 5 is devoted to the proof of a result on edge-coloured multigraphs that affirms asymptotically an algebraic question by Grinblat and is a partial result towards a conjecture by Aharoni and Berger. This chapter is based on joint work with Dennis Clemens and Alexey Pokrovskiy [56] and with Dennis Clemens [53].
Finally in Chapter 6 we prove an asymptotic estimate of the expected number of spanning trees in a labelled connected series-parallel graph chosen uniformly at random. Furthermore, we obtain analogue results for subfamilies of series-parallel graphs such as 2-trees and con-nected series-parallel graphs with fixed excess. This is based on joint work with Juanjo Ru´e [75].
Tools and notation 2
2.1 Definitions
In this section we introduce the basic definitions that are frequently used throughout the present thesis. The terminologies that we need with regard to the regularity method and analytic combinatorics are introduced in Section 2.2 and Section 2.3, respectively. Other definitions, which occur rarely or are used in only certain parts of the thesis, are deferred to the places where they are needed. For all elementary graph theoretic concepts not defined in this section we refer the reader to e.g. [62].
2.1.1 General notions
Throughout the thesis log denotes the natural logarithm, whereas we write log2 for the loga-rithm to the base 2. For a positive integernwe define [n] :={1, . . . , n}and for reals a, b >0 we writex =a±b if and only ifa−b ≤x ≤a+b. For the sake of brevity, when we write x >0 we always assumexto be real and when we writex≥nwithnbeing a positive integer, we always assume thatx is an integer, unless stated otherwise. Given a setS and a positive integerr≤ |S|we let Sr
denote the set{S0 ⊆S :|S0|=r}. For a sequenceA= (a1, . . . , ak) we writea∈ Aif there exists an indexi∈[k] witha=ai.
Given a bivariate functionA(x, y), we denote the partial derivative ofA(x, y) with respect tox and y by Ax(x, y) and Ay(x, y), respectively. However, we will usually use the notation A0(x, y) :=Ax(x, y) for the derivative of A(x, y) with respect to the first variable.
To express asymptotic behaviours we use the standard Landau notation. In particular, given two functions f, g : N → R\ {0} we write f = o(g) if limn→∞|f(n)/g(n)| = 0 and f = ω(g) if limn→∞|g(n)/f(n)| = 0. If there exist constants C >0 and n0 ∈ N such that
|f(n)| ≤C|g(n)|for every n≥n0, we use f =O(g) and if there existc > 0 andn0 ∈N such that |f(n)| ≥ c|g(n)|, we write f = Ω(g). If f = Ω(g) and f = O(g), we use the notation f = Θ(g). Finally, if limn→∞f(n)/g(n) = 1, we say that f and g are asymptotically equal and write f ∼ g. In all cases, for the sake of simplicity, we may replace f by f(n) or g by g(n), for instance we may write f =O(n).
By Bin(n, p) we denote the binomial distribution with parametersnand p.
25
2.1.2 Graphs and multigraphs
Most of the graph-theoretic terminology that we use is standard and follows [62]. All graphs and hypergraphs in this thesis are finite and undirected. Graphs and hypergraphs are always simple, while multigraphs may have multiple edges and loops, unless otherwise specified.
A graph G is a pair (V, E) of sets V and E with E ⊆ V2
, where V is called vertex set and E edge set of G. We say that vertices x, y ∈ V are adjacent if{x, y} ∈E(G), and say that a vertex x∈ V and an edge e∈E are incident ifx ∈e. Two distinct edges are called adjacent if their intersection is nonempty. Given a graphG, we useV(G) andE(G) to refer to its vertex and edge set, respectively.
LetG= (V, E) be a graph and letA, B⊆V be disjoint. We denote the set of edges inAby EG(A) :={e∈E : e⊆A}and the set of those between A and B by EG(A, B) :=
{x, y} ∈ E : x∈Aand y ∈B . The cardinalities of these sets are denoted byeG(A) :=|EG(A)|and eG(A, B) :=|EG(A, B)|. For a vertexx∈V we writeNG(x) :={y∈V : {x, y} ∈E}for the neighbourhood ofx inGand NG(x, A) :=NG(x)∩Afor the neighbourhood ofxrestricted to A. Given verticesx1, . . . , xk∈V we denote thejoint neighbourhood of x1, . . . , xk restricted toA by NG(x1, . . . , xk;A) =T
i∈[k]NG(xi, A).
Thedegree of a vertexx∈V is the size of its neighbourhood and is denoted by degG(x) :=
|NG(x)|. Similarly, we use the notation degG(x, A) :=|NG(x, A)|and degG(x1, . . . , xk;A) :=
|NG(x1, . . . , xk;A)| for the degree of x restricted to A in G and the size of the joint neigh-bourhood of x1, . . . , xk restricted to A in G, respectively. For short notation we may omit the subscript Gwhen there is no risk of confusion.
Theminimum andmaximum degree of Gare denoted byδ(G) and ∆(G), respectively. If δ(G) = ∆(G), thenGis calledregular and for the special case δ(G) = ∆(G) = 3, we say that G iscubic.
Given a graphG= (V, E), we say that a graphH= (V0, E0) is asubgraph ofGifV0 ⊆V and E0 ⊆ E. If H is a subgraph of G we write H ⊆ G. A subgraph H = (V0, E0) ⊆ G is spanning if V0 = V and induced if for all edges e ∈ E with e ⊆ V0 it holds that e ∈ E0. We write G[V0] for the induced subgraph of G on the vertex set V0. A set W ⊆ V is called independent if the edge set of G[W] is empty. Moreover, for a set A ⊆ V we write G−A:=G[V \A].
A graph homomorphism or simply homomorphism from a graph H to a graph G is a mapping h :V(H) → V(G) such that for every {x, y} ∈ E(H) it holds that {h(x), h(y)} ∈ E(G). Two graphs H and H0 are isomorphic if there exists a bijective homomorphism h from H toG such that the inverse function h−1 is again a homomorphism. In this case we also say that H is acopy of H0. IfGhas a subgraph that is isomorphic toH, we say thatG contains a copy of H or that there is an embedding of H onto G. When it is clear from the context we simply say that GcontainsH (as a subgraph). A graphGis called universal for a class of graphs HifG contains all graphs fromH as subgraphs.
We say that a graph G is connected if for all vertices x, y ∈ V(G) there is a path in G with endpoints x and y. A maximal connected subgraph of G is a connected component or simply a component of G. A graph G is calledk-connected if|V(G)| ≥k and if one cannot disconnect G by deleting (k−1) vertices of G. A graph that does not contain any cycles is called aforest oracyclic. Connected acyclic graphs aretrees.
A Hamilton cycle in a graph G is a spanning subgraph of G that is a cycle. A graph G is said to be Hamiltonian ifG contains a Hamilton cycle. Given a graph G, the distance
2.1. Definitions 27
distG(x, y) between two vertices x, y∈V(G) is the length of a shortest path in Gconnecting them. Thek-th power of a graph Gis the graph on the vertex set V(G) such that the edge set consists of all sets{x, y}for which x, y∈V(G) and distG(x, y)≤k. The second power of a graphGis also called square of G.
Given a graphH, anH-factor of a graphGis a subgraph ofGconsisting of disjoint unions of copies of H. An H-factor is called perfect if it is spanning. The edge set of a K2-factor is called matching and the edge set of a perfect K2-factor perfect matching. The size of a matchingM is defined as the number of edges of M. A vertexv is said to be saturated by a matchingM if there is an edgee∈M withv∈e.
A graph property is called monotone increasing if the property is preserved under edge and vertex insertions. We say that a graph G is edge-maximal with respect to some graph property P if G has property P but adding any edge to G produces a graph that does no longer have propertyP.
A graph H is aminor of a graphGifH can be obtained from Gby a sequence of vertex and edge deletions as well as edge contractions. We say that a familyH0 of graphs isH-minor free for a family H of graphs if no graph ofH0 contains a graph from H as a minor.
Outerplanar graphs are the family of{K4, K3,2}-minor free graphs,series-parallel graphs are those that exclude K4 as a minor, and a graph is called planar if it neither contains K5 norK3,3 as a minor. Edge-maximal series-parallel graphs are called 2-trees. Equivalent definitions of series-parallel graphs and 2-trees are discussed in Chapters 1 and 6.
Apartition of a graphGis a partition of its vertex setV(G) into disjoint sets. These sets are referred to aspartition classes.
A vertex-colouring of a graph is a map from the vertex set of a graph to a set of colours.
Anedge-colouring is defined analogously. Acolouring is said to beproper if adjacent vertices or edges do not receive the same colour. A graph is said to bek-colourable if there exists a proper colouring of its vertex set withk colours. A colour class of a colouring is a maximal set of vertices or edges of the same colour.
A graphGis calledk-partite, if Gisk-colourable. Forr= 2 we say that Gisbipartite. If Gis bipartite we may writeG= (A∪B, E) to indicate that Aand B are partition classes of a bipartition ofG.
We letKndenote the complete graph onnvertices andKn,m the complete bipartite graph with partition classes of sizenand m. The complete graph K3 is also calledtriangle.
Given a graph G= (V, E), a labelling of its vertex set is a bijective function from V to [|V|]. We define alabelled graph to be a pair (V, E) equipped with a labelling ofV. Informally, a labelled graph is a graph whose vertices bear distinct labels.
Thebandwidth of a graph, denoted by bw(G), is defined as the minimum positive integer bsuch that there is a labelling of V(G) such that|i−j| ≤bfor every edge {i, j} ∈E(G).
Formally, a multigraph is a pair (V, E) of sets V and E with a map E → V ∪(V ×V), i.e. every edge is mapped to one or two vertices. One can think of a multigraph as a graph with multiple edges and loops. For any vertices x, y ∈ V, the number of edges that are mapped to{x, y} is called multiplicity of{x, y}.
2.1.3 Hypergraphs
For every integer r ≥ 3, an r-uniform hypergraph is a pair (V, E) of a vertex set V and an edge set E with E ⊆ Vr
. The elements of E are referred to as edges or hyperedges.
Given a hypergraph H, as in the setting of graphs, we use V(H) and E(H) to refer to its vertex and edge set, respectively. Also, the definition of subgraphs generalises naturally to subhypergraphs of hypergraphs.
Given anr-uniform hypergraph H= (V, E) and a setS ⊆V with|S| ≤r−1, thedegree ofS is defined as degH(S) :=|{e∈E :S⊆e}|. Theminimum d-degree δd(H) ofH is defined asδd(H) := min{degH(S) : S ⊆V,|S| =d}. We call degH({x}) the vertex degree of x and δ1(H) the minimum vertex degree of H and use degH(x) := degH({x}) and δ(H) := δ1(H) for short notation. We may omit the subscriptH whenever there is no risk of confusion.
Anr-uniform hypergraphH is r-partite if its vertex set can be partitioned intor disjoint sets such that each hyperedge contains exactly one element from each of thersets. Amatching of a hypergraphH is a set of pairwise disjoint hyperedges. A matchingM ofH is perfect if every vertex ofH is contained in a hyperedge of M.
Aweak cycle is an alternating sequence (v1, e1, v2, . . . , vk, ek) of distinct verticesv1, . . . , vk and hyperedgese1, . . . , ek such that{v1, vk} ⊆ek and{vi, vi+1} ⊆ei for everyi∈[k−1]. A weak cycle is calledBerge cycle if all its hyperedges are distinct. IfP = (v1, e1, v2, . . . , vn, en) is a weak cycle or a Berge cycle in a hypergraph H on n vertices, then P is called weak Hamilton cycle orHamilton Berge cycle ofH, respectively. A hypergraphH is called Berge-acyclic ifH does not contain any Berge cycle as a subgraph.
Given anr-uniform hypergraphH = (V, E) and setsA1, A2, . . . , A` ⊆V as well as positive integersr1, r2, . . . , r` with P
i∈[`]ri =r, we write EH A(r11), A(r22), . . . , A(r` `)
:=
e∈E(H) : ∃(v1, v2, . . . , vr)∈Ar11 ×Ar22 ×. . .×Ar`` withe={v1, v2, . . . , vr} .
Moreover, we define eH A(r11), A(r22), . . . , A(r` `)
:= |EH A(r11), A(r22), . . . , A(r` `)
|. Again, we may omit the subscriptH whenever it is clear from the context.
Finally, we denote the complete r-uniform hypergraph onnvertices byKn(r). 2.1.4 Random graphs and hypergraphs
The random graph model that we consider is the Erd˝os-R´enyi modelG(n, p), which is defined on the vertex set [n] where each pair of vertices forms an edge randomly and independently with probabilityp=p(n). The generalisation ofG(n, p) to hypergraphs is defined as follows.
For every r ≥ 3 we denote by H(r)(n, p) the random r-uniform hypergraph model on the vertex set [n], where each set of r vertices forms an edge randomly and independently with probabilityp=p(n).
Given a functionp:N→[0,1], we say thatG(n, p) has a graph propertyP asymptotically almost surely (or a.a.s. for short) if limn→∞P[G(n, p) ∈ P] = 1. Furthermore, the threshold for an increasing propertyP is defined as a sequence ˆp= ˆp(n) such that
nlim→∞P[G(n, p)∈ P] =
(0 ifp=o(ˆp), 1 ifp=ω(ˆp).