A classical problem, due to Gerencsér and Gyárfás from 1967, asks how large a monochromatic connected component can we guarantee in any r-edge colouring of $K_n$? We consider how big a connected component we can guarantee in any r-edge colouring of $K_n$ if we allow ourselves to use up to s colours. This is actually an instance of a more general question of Bollobás from about 20 years ago which asks for a k-connected subgraph in the same setting. We complete the picture in terms of the approximate behaviour of the answer by determining it up to a logarithmic term, provided n is large enough. We obtain more precise results for certain regimes which solve a problem of Liu, Morris and Prince from 2007, as well as disprove a conjecture they pose in a strong form.

We also consider a generalisation in a similar direction of a question first considered by Erdős and Rényi in 1956, who considered given n and m, what is the smallest number of m-cliques which can cover all edges of $K_n$? This problem is essentially equivalent to the question of what is the minimum number of vertices that are certain to be incident to at least one edge of some colour in any r-edge colouring of $K_n$. We consider what happens if we allow ourselves to use up to s colours. We obtain a more complete understanding of the answer to this question for large n, in particular, determining it up to a constant factor for all $1\le s \le r$, as well as obtaining much more precise results for various ranges including the correct asymptotics for essentially the whole range.

A digraph group is a group defined by non-empty presentation with the property that each relator is of the form $R(x, y)$, where x and y are distinct generators and $R(\cdot , \cdot )$ is determined by some fixed cyclically reduced word $R(a, b)$ that involves both a and b. Associated with each such presentation is a digraph whose vertices correspond to the generators and whose arcs correspond to the relators. In this article, we consider digraph groups for strong digraphs that are digon-free and triangle-free. We classify when the digraph group is finite and show that in these cases it is cyclic, giving its order. We apply this result to the Cayley digraph of the generalized quaternion group, to circulant digraphs, and to Cartesian and direct products of strong digraphs.

The study of threshold functions has a long history in random graph theory. It is known that the thresholds for minimum degree k, k-connectivity, as well as k-robustness coincide for a binomial random graph. In this paper we consider an inhomogeneous random graph model, which is obtained by including each possible edge independently with an individual probability. Based on an intuitive concept of neighborhood density, we show two sufficient conditions guaranteeing k-connectivity and k-robustness, respectively, which are asymptotically equivalent. Our framework sheds some light on extending uniform threshold values in homogeneous random graphs to threshold landscapes in inhomogeneous random graphs.

An edge-coloured graph $G$ is called properly connected if any two vertices are connected by a properly coloured path. The proper connection number, $pc(G)$, of a graph $G$, is the smallest number of colours that are needed to colour $G$ such that it is properly connected. Let $\unicode[STIX]{x1D6FF}(n)$ denote the minimum value such that $pc(G)=2$ for any 2-connected incomplete graph $G$ of order $n$ with minimum degree at least $\unicode[STIX]{x1D6FF}(n)$. Brause et al. [‘Minimum degree conditions for the proper connection number of graphs’, Graphs Combin.33 (2017), 833–843] showed that $\unicode[STIX]{x1D6FF}(n)>n/42$. In this note, we show that $\unicode[STIX]{x1D6FF}(n)>n/36$.

If we pick n random points uniformly in $[0,1]^d$ and connect each point to its $c_d \log{n}$ nearest neighbors, where $d\ge 2$ is the dimension and $c_d$ is a constant depending on the dimension, then it is well known that the graph is connected with high probability. We prove that it suffices to connect every point to $ c_{d,1} \log{\log{n}}$ points chosen randomly among its $ c_{d,2} \log{n}$ nearest neighbors to ensure a giant component of size $n - o(n)$ with high probability. This construction yields a much sparser random graph with $\sim n \log\log{n}$ instead of $\sim n \log{n}$ edges that has comparable connectivity properties. This result has non-trivial implications for problems in data science where an affinity matrix is constructed: instead of connecting each point to its k nearest neighbors, one can often pick $k'\ll k$ random points out of the k nearest neighbors and only connect to those without sacrificing quality of results. This approach can simplify and accelerate computation; we illustrate this with experimental results in spectral clustering of large-scale datasets.

In the framework of coupled cell systems, a coupled cell network describes graphically the dynamical dependencies between individual dynamical systems, the cells. The fundamental network of a network reveals the hidden symmetries of that network. Subspaces defined by equalities of coordinates which are flow-invariant for any coupled cell system consistent with a network structure are called the network synchrony subspaces. Moreover, for every synchrony subspace, each network admissible system restricted to that subspace is a dynamical system consistent with a smaller network called a quotient network. We characterize networks such that: the network is a subnetwork of its fundamental network, and the network is a fundamental network. Moreover, we prove that the fundamental network construction preserves the quotient relation and it transforms the subnetwork relation into the quotient relation. The size of cycles in a network and the distance of a cell to a cycle are two important properties concerning the description of the network architecture. In this paper, we relate these two architectural properties in a network and its fundamental network.

A class of graphs is bridge-addable if given a graph $G$ in the class, any graph obtained by adding an edge between two connected components of $G$ is also in the class. The authors recently proved a conjecture of McDiarmid, Steger, and Welsh stating that if ${\mathcal{G}}$ is bridge-addable and $G_{n}$ is a uniform $n$-vertex graph from ${\mathcal{G}}$, then $G_{n}$ is connected with probability at least $(1+o_{n}(1))e^{-1/2}$. The constant $e^{-1/2}$ is best possible, since it is reached for the class of all forests.

In this paper, we prove a form of uniqueness in this statement: if ${\mathcal{G}}$ is a bridge-addable class and the random graph $G_{n}$ is connected with probability close to $e^{-1/2}$, then $G_{n}$ is asymptotically close to a uniform $n$-vertex random forest in a local sense. For example, if the probability converges to $e^{-1/2}$, then $G_{n}$ converges in the sense of Benjamini–Schramm to the uniformly infinite random forest $F_{\infty }$. This result is reminiscent of so-called “stability results” in extremal graph theory, the difference being that here the stable extremum is not a graph but a graph class.

In this paper we consider two natural notions of connectivity for hypergraphs: weak and strong. We prove that the strong vertex connectivity of a connected hypergraph is bounded by its weak edge connectivity, thereby extending a theorem of Whitney from graphs to hypergraphs. We find that, while determining a minimum weak vertex cut can be done in polynomial time and is equivalent to finding a minimum vertex cut in the 2-section of the hypergraph in question, determining a minimum strong vertex cut is NP-hard for general hypergraphs. Moreover, the problem of finding minimum strong vertex cuts remains NP-hard when restricted to hypergraphs with maximum edge size at most 3. We also discuss the relationship between strong vertex connectivity and the minimum transversal problem for hypergraphs, showing that there are classes of hypergraphs for which one of the problems is NP-hard, while the other can be solved in polynomial time.

We identify the asymptotic probability of a configuration model CMn(d) producing a connected graph within its critical window for connectivity that is identified by the number of vertices of degree 1 and 2, as well as the expected degree. In this window, the probability that the graph is connected converges to a non-trivial value, and the size of the complement of the giant component weakly converges to a finite random variable. Under a finite second moment condition we also derive the asymptotics of the connectivity probability conditioned on simplicity, from which follows the asymptotic number of simple connected graphs with a prescribed degree sequence.

Computer or communication networks are so designed that they do not easily get disrupted under external attack. Moreover, they are easily reconstructed when they do get disrupted. These desirable properties of networks can be measured by various parameters, such as connectivity, toughness and scattering number. Among these parameters, the isolated scattering number is a comparatively better parameter to measure the vulnerability of networks. In this paper we first prove that for split graphs, this number can be computed in polynomial time. Then we determine the isolated scattering number of the Cartesian product and the Kronecker product of special graphs and special permutation graphs.

Consider a bipartite random geometric graph on the union of two independent homogeneous Poisson point processes in d-space, with distance parameter r and intensities λ and μ. We show for d ≥ 2 that if λ is supercritical for the one-type random geometric graph with distance parameter 2r, there exists μ such that (λ, μ) is supercritical (this was previously known for d = 2). For d = 2, we also consider the restriction of this graph to points in the unit square. Taking μ = τ λ for fixed τ, we give a strong law of large numbers as λ → ∞ for the connectivity threshold of this graph.

Let $\mathcal{A}$ be a line arrangement in the complex projective plane ${{\mathbb{P}}^{2}}$ and let $M$ be its complement. A rank one local system $\mathcal{L}$ on $M$ is admissible if, roughly speaking, the cohomology groups ${{H}^{m}}\left( M,\,\mathcal{L} \right)$ can be computed directly from the cohomology algebra ${{H}^{*}}\left( M,\,\mathbb{C} \right)$. In this work, we give a sufficient condition for the admissibility of all rank one local systems on $M$. As a result, we obtain some properties of the characteristic variety ${{\mathcal{V}}_{1}}\left( M \right)$ and the Resonance variety ${{\mathcal{R}}_{1}}\left( M \right)$.

Given two independent Poisson point processes Φ(1), Φ(2) in , the AB Poisson Boolean model is the graph with the points of Φ(1) as vertices and with edges between any pair of points for which the intersection of balls of radius 2r centered at these points contains at least one point of Φ(2). This is a generalization of the AB percolation model on discrete lattices. We show the existence of percolation for all d ≥ 2 and derive bounds for a critical intensity. We also provide a characterization for this critical intensity when d = 2. To study the connectivity problem, we consider independent Poisson point processes of intensities n and τn in the unit cube. The AB random geometric graph is defined as above but with balls of radius r. We derive a weak law result for the largest nearest-neighbor distance and almost-sure asymptotic bounds for the connectivity threshold.

In a cubic multigraph certain restrictions on the paths are made to define what is called a railway. On the tracks in the railway (edges in the multigraph) an equivalence relation is defined. The number of equivalence classes induced by this relation is investigated for a random railway achieved from a random cubic multigraph, and the asymptotic distribution of this number is derived as the number of vertices tends to infinity.

A separator of a connected graph $G$ is a set of vertices whose removal disconnects $G$. In this paper we give various conditions for a separator to contain a minimal one. In particular we prove that every separator of a connected graph that has no thick end, or which is of bounded degree, contains a minimal separator.