Related papers: Typical large graphs with given edge and triangle …
For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…
We describe the structure of connected graphs with the minimum and maximum average distance, radius, diameter, betweenness centrality, efficiency and resistance distance, given their order and size. We find tight bounds on these graph…
We determine the minimal density of triangles in a tripartite graph with prescribed edge densities. This extends a previous result of Bondy, Shen, Thomass\'e and Thomassen characterizing those edge densities guaranteeing the existence of a…
It is known that Paley graphs of square order have the strict-EKR property, that is, all maximum cliques are canonical cliques. Peisert-type graphs are natural generalizations of Paley graphs and some of them also have the strict-EKR…
We characterise the form of all simple, finite graphs for which the girth of the graph is equal to the circumference of the graph. We apply this to prove a bound on the number of edges in such a graph.
A good edge-labelling of a simple graph is a labelling of its edges with real numbers such that, for any ordered pair of vertices (u,v), there is at most one nondecreasing path from u to v. Say a graph is good if it admits a good…
We prove that the asymptotic entropy of large simple graphs, as a function of fixed edge and triangle densities, is nondifferentiable along a certain curve.
A key concept for many graph layout algorithms is planarity, a graph property that allows to draw vertices and edges crossing-free in the plane. Important is the generalization to $k$-planar graphs, which can be drawn in the plane with at…
We introduce and analyse ensembles of 2-regular random graphs with a tuneable distribution of short cycles. The phenomenology of these graphs depends critically on the scaling of the ensembles' control parameters relative to the number of…
We show that for an infinitely many natural numbers $k$ there are $k$-uniform hypergraphs which admit a `rescaling phenomenon' as described in [9]. More precisely, let $\mathcal{A}(k,I, n)$ denote the class of $k$-graphs on $n$ vertices in…
A $d$-regular graph on $n$ nodes has at most $T_{\max} = \frac{n}{3} \tbinom{d}{2}$ triangles. We compute the leading asymptotics of the probability that a large random $d$-regular graph has at least $c \cdot T_{\max}$ triangles, and…
Large graphs are sometimes studied through their degree sequences (power law or regular graphs). We study graphs that are uniformly chosen with a given degree sequence. Under mild conditions, it is shown that sequences of such graphs have…
In this paper we consider the Erd\H{o}s-R\'enyi random graph in the sparse regime in the limit as the number of vertices $n$ tends to infinity. We are interested in what this graph looks like when it contains many triangles, in two…
We prove the well-known Brown-Erd\H{o}s-S\'os Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is…
A tight $\ell$-cycle minus an edge $C_\ell^-$ is the $3$-graph on the vertex set $[\ell]$, where any three consecutive vertices in the string $123\ldots\ell 1$ form an edge. We show that for every $\ell\ge 5$, $\ell$ not divisible by $3$,…
We define the limiting density of a minor-closed family of simple graphs F to be the smallest number k such that every n-vertex graph in F has at most kn(1+o(1)) edges, and we investigate the set of numbers that can be limiting densities.…
For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…
The unconstrained exponential family of random graphs assumes no prior knowledge of the graph before sampling, but it is natural to consider situations where partial information about the graph is known, for example the total number of…
A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…
A \emph{$k$-planar graph} is a graph that can be drawn in the plane such that every edge is crossed at most $k$ times. For $k \leq 4$, Pach and T\'oth proved a bound of $(k+3)(n-2)$ on the total number of edges of a $k$-planar graph, which…