Related papers: Tricolor percolation and random paths in 3D
The flip graph of triangulations has as vertices all triangulations of a convex $n$-gon, and an edge between any two triangulations that differ in exactly one edge. An $r$-rainbow cycle in this graph is a cycle in which every inner edge of…
\emph{Full-bond percolation} with parameter $p$ is the process in which, given a graph, for every edge independently, we delete the edge with probability $1-p$. Bond percolation is motivated by problems in mathematical physics and it is…
Techniques of `dynamic renormalization', developed earlier for undirected percolation and the contact model, are adapted to the setting of directed percolation, thereby obtaining solutions of several problems for directed percolation on…
In a geometric inhomogeneous random graph vertices are given by the points of a Poisson process and are equipped with independent weights following a heavy tailed distribution. Any pair of distinct vertices is independently forming an edge…
A famous conjecture of Caccetta and H\"aggkvist is that in a digraph on $n$ vertices and minimum out-degree at least $\frac{n}{r}$ there is a directed cycle of length $r$ or less. We consider the following generalization: in an undirected…
Classical bond percolation theory studies the conditions for a given point in a random graph to be connected to infinity, or "escape" to infinity, via a sequence of random edges. In this work, we present a higher-dimensional generalization…
In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a…
Settling a first case of a conjecture of M. Kahle on the homology of the clique complex of the random graph $G=G_{n,p}$, we show, roughly speaking, that (with high probability) the triangles of $G$ span its cycle space whenever each of its…
The finite-size scaling behaviour for percolation and conduction is studied in two-dimensional triangular-shaped random resistor networks at the percolation threshold. The numerical simulations are performed using an efficient star-triangle…
Given independent normally distributed points A,B,C,D in Euclidean 3-space, let Q denote the plane determined by A,B,C and D^ denote the orthogonal projection of D onto Q. The probability that the tetrahedron ABCD is acute remains…
Let G be a planar triangle-free graph and let C be a cycle in G of length at most 8. We characterize all situations where a 3-coloring of C does not extend to a proper 3-coloring of the whole graph.
In \emph{$k$-bootstrap percolation}, we fix $p\in (0,1)$, an integer $k$, and a plane graph $G$. Initially, we infect each face of $G$ independently with probability $p$. Infected faces remain infected forever, and if a healthy (uninfected)…
A walk $W$ in edge-colored graphs is called properly colored (PC) if every pair of consecutive edges in $W$ is of different color. We introduce and study five types of PC acyclicity in edge-colored graphs such that graphs of PC acyclicity…
We give a linear-time algorithm to decide 3-colorability (and find a 3-coloring, if it exists) of quadrangulations of a fixed surface. The algorithm also allows to prescribe the coloring for a bounded number of vertices.
A classical result of Corr\'adi and Hajnal states that every graph $G$ on $n$ vertices with $n\in 3\mathbb{N}$ and $\delta(G) \ge 2n/3$ contains a perfect triangle-tiling, i.e.,\ a spanning set of vertex-disjoint triangles. We explore a…
We study site percolation on Angel & Schramm's uniform infinite planar triangulation. We compute several critical and near-critical exponents, and describe the scaling limit of the boundary of large percolation clusters in all regimes…
We show how to adjust a very nice coupling argument due to McDiarmid in order to prove/reprove in a novel way results concerning Hamilton cycles in various models of random graph and hypergraphs. In particular, we firstly show that for…
We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either…
We introduce and solve a two-matrix model for the tri-coloring problem of the vertices of a random triangulation. We present three different solutions: (i) by orthogonal polynomial techniques (ii) by use of a discrete Hirota bilinear…
A well-known result by Graham in Euclidean Ramsey Theory states that, for every positive real number $A$, every coloring of the plane with finite number of colors contains a monochromatic triangle of area $A$. We consider canonical versions…