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We consider a generalised model of a random simplicial complex, which arises from a random hypergraph. Our model is generated by taking the downward-closure of a non-uniform binomial random hypergraph, in which for each $k$, each set of $k+1$ vertices forms an edge with some probability $p_k$ independently. As a special case, this contains an extensively studied model of a (uniform) random simplicial complex, introduced by Meshulam and Wallach [Random Structures & Algorithms 34 (2009), no. 3, pp. 408-417]. We consider a higher-dimensional notion of connectedness on this new model according to the vanishing of cohomology groups over an arbitrary abelian group $R$. We prove that this notion of connectedness displays a phase transition and determine the threshold. We also prove a hitting time result for a natural process interpretation, in which simplices and their downward-closure are added one by one. In addition, we determine the asymptotic behaviour of cohomology groups inside the critical window around the time of the phase transition., Comment: 57 pages

We consider $k$-dimensional random simplicial complexes that are generated from the binomial random $(k+1)$-uniform hypergraph by taking the downward-closure, where $k\geq 2$. For each $1\leq j \leq k-1$, we determine when all cohomology groups with coefficients in $\mathbb{F}_2$ from dimension one up to $j$ vanish and the zero-th cohomology group is isomorphic to $\mathbb{F}_2$. This property is not deterministically monotone for this model of random complexes, but nevertheless we show that it has a single sharp threshold. Moreover we prove a hitting time result, relating the vanishing of these cohomology groups to the disappearance of the last minimal obstruction. We also study the asymptotic distribution of the dimension of the $j$-th cohomology group inside the critical window. As a corollary, we deduce a hitting time result for a different model of random simplicial complexes introduced in [Linial and Meshulam, Combinatorica, 2006], a result which was previously only known for dimension two [Kahle and Pittel, Random Structures Algorithms, 2016]., Comment: 35 pages

For integers $g,m \geq 0$ and $n>0$, let $S_{g}(n,m)$ denote the graph taken uniformly at random from the set of all graphs on $\{1,2, \ldots, n\}$ with exactly $m=m(n)$ edges and with genus at most $g$. We use counting arguments to investigate the components, subgraphs, maximum degree, and largest face size of $S_{g}(n,m)$, finding that there is often different asymptotic behaviour depending on the ratio $\frac{m}{n}$. In our main results, we show that the probability that $S_{g}(n,m)$ contains any given non-planar component converges to $0$ as $n \to \infty$ for all $m(n)$; the probability that $S_{g}(n,m)$ contains a copy of any given planar graph converges to $1$ as $n \to \infty$ if $\liminf \frac{m}{n} > 1$; the maximum degree of $S_{g}(n,m)$ is $\Theta (\ln n)$ with high probability if $\liminf \frac{m}{n} > 1$; and the largest face size of $S_{g}(n,m)$ has a threshold around $\frac{m}{n}=1$ where it changes from $\Theta (n)$ to $\Theta (\ln n)$ with high probability., Comment: 35 pages

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We prove that the component structure of a graph chosen uniformly at random from the class $\mathcal{S}_g(n,m)$ of all graphs on vertex set $[n]=\{1,\dotsc,n\}$ with $m$ edges embeddable on $\mathbb{S}_g$ features two phase transitions. The first phase transition mirrors the classical phase transition in the Erd\H{o}s--R\'enyi random graph $G(n,m)$ chosen uniformly at random from all graphs with vertex set $[n]$ and $m$ edges. It takes place at $m=\frac{n}{2}+O(n^{2/3})$, when a unique largest component, the so-called \emph{giant component}, emerges. The second phase transition occurs at $m = n+O(n^{3/5})$, when the giant component covers almost all vertices of the graph. This kind of phenomenon is strikingly different from $G(n,m)$ and has only been observed for graphs on surfaces. Moreover, we derive an asymptotic estimation of the number of graphs in $\mathcal{S}_g(n,m)$ throughout the regimes of these two phase transitions., Comment: 47 pages, 1 figure. An extended abstract of this paper has been published in the Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb17), Electronic Notes in Discrete Mathematics 61:687--693, 2017

We consider simplicial complexes that are generated from the binomial random 3-uniform hypergraph by taking the downward-closure. We determine when this simplicial complex is homologically connected, meaning that its zero-th and first homology groups with coefficients in $\mathbb{F}_2$ vanish. Although this is not intrinsically a monotone property, we show that it nevertheless has a single sharp threshold, and indeed prove a hitting time result relating the connectedness to the disappearance of the last minimal obstruction., Comment: 21 pages

Let $\mathbb{S}_g$ be the orientable surface of genus $g$. We show that the number of vertex-labelled cubic multigraphs embeddable on $\mathbb{S}_g$ with $2n$ vertices is asymptotically $c_g n^{5(g-1)/2-1}\gamma^{2n}(2n)!$, where $\gamma$ is an algebraic constant and $c_g$ is a constant depending only on the genus $g$. We also derive an analogous result for simple cubic graphs and weighted cubic multigraphs. Additionally we prove that a typical cubic multigraph embeddable on $\mathbb{S}_g$, $g\ge 1$, has exactly one non-planar component., Comment: 50 pages. An extended abstract of this paper has been published in the Proceedings of the European Conference on Combinatorics, Graph Theory and Applications (EuroComb15), Electronic Notes in Discrete Mathematics (2015), 603--610

We give a full characterisation of the symmetries of unlabelled triangulations and derive a constructive decomposition of unlabelled triangulations depending on their symmetries. As an application of these results we can deduce a complete enumerative description of unlabelled cubic planar graphs., Comment: 37 pages, 21 figures. An extended abstract of this paper has been accepted for the proceedings of EUROCOMB 2015

This paper is the last part of a comprehensive survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. The first two parts of the survey together provide a suitable entry point to this field for new readers; they are available in combined form in arXiv:0912.4213. The topological approach indicated above has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. While the first part of this survey introduces the theory as such and the second part is devoted to those applications, this third part looks at the theory from an algebraic-topological point of view. The results surveyed here include both a combinatorial description of the fundamental group of a locally finite graph with ends and the homology aspects of this space., Comment: 15 pages, 7 figures; corrected typo

We prove a strong version of the Max-Flow Min-Cut theorem for countable networks, namely that in every such network there exist a flow and a cut that are "orthogonal" to each other, in the sense that the flow saturates the cut and is zero on the reverse cut. If the network does not contain infinite trails then this flow can be chosen to be mundane, i.e. to be a sum of flows along finite paths. We show that in the presence of infinite trails there may be no orthogonal pair of a cut and a mundane flow. We finally show that for locally finite networks there is an orthogonal pair of a cut and a flow that satisfies Kirchhoff's first law also for ends., Comment: 19 pages, to be published in Journal of Combinatorial Theory, Series B

Given an assignment of weights w to the edges of a graph G, a matching M in G is called strongly w-maximal if for any matching N the sum of weights of the edges in N\M is at most the sum of weights of the edges in M\N. We prove that if w assumes only finitely many values all of which are rational then G has a strongly w-maximal matching., Comment: 17 pages

We show that the topological space of any infinite graph and its ends is normal. In particular, end spaces themselves are normal., Comment: 8 pages

We prove that the topological cycle space C(G) of a locally finite graph G is generated by its geodetic topological circles. We further show that, although the finite cycles of G generate C(G), its finite geodetic cycles need not generate C(G)., Comment: 17

Let H be an r-partite r-graph, all of whose sides have the same size n. Suppose that there exist two sides of H, each satisfying the following condition: the degree of each legal (r-1)-tuple contained in the complement of this side is strictly larger than n/2. We prove that under this condition H must have a perfect matching. This answers a question of Kuhn and Osthus., Comment: 6 pages

Two non-isomorphic graphs are twins if each is isomorphic to a subgraph of the other. We prove that a rayless graph has either infinitely many twins or none., Comment: 8 pages

We propose a homology theory for locally compact spaces with ends in which the ends play a special role. The approach is motivated by results for graphs with ends, where it has been highly successful. But it was unclear how the original graph-theoretical definition could be captured in the usual language for homology theories, so as to make it applicable to more general spaces. In this paper we provide such a general topological framework: we define a homology theory which satisfies the usual axioms, but which maintains the special role for ends that has made this homology work so well for graphs., Comment: 23 pages; final version, to appear in Topology and its Applications

We characterize the fundamental group of a locally finite graph G with ends combinatorially, as a group of infinite words. Our characterization gives rise to a canonical embedding of this group in the inverse limit of the (free) fundamental groups of the finite subgraphs of G., Comment: 38 pages. This is an extended version of the paper "The fundamental group of a locally finite graph with ends" by the same authors. It differs from that paper only in that it offers proofs for Lemmas 2, 4, 6, 7, 8, 9 and 20, and a longer example in Section 5

We show that the topological cycle space of a locally finite graph is a canonical quotient of the first singular homology group of its Freudenthal compactification, and we characterize the graphs for which the two coincide. We construct a new singular-type homology for non-compact spaces with ends, which in dimension~1 captures precisely the topological cycle space of graphs but works in any dimension., Comment: 30 pages. This is an extended version of the paper "The homology of a locally finite graph with ends" (to appear in Combinatorica) by the same authors. It differs from that paper only in that it offers proofs for Lemmas 3, 4 and 10, as well as a new footnote in Section 6

We (re-)prove that in every 3-edge-coloured tournament in which no vertex is incident with all colours there is either a cyclic rainbow triangle or a vertex dominating every other vertex monochromatically., Comment: We found out that our main result had appeared before (in [3]), and updated the paper to add this information

We prove that every rayless graph has an unfriendly partition., Comment: Final, slightly modified, version. To appear in combinatorica