Related papers: Rainbow Thresholds
Resolving a recent problem of Bell, Frieze, and Marbach, we establish both the threshold result of Frankston--Kahn--Narayanan--Park, and its strengthening by Spiro, in the rainbow setting. This has applications to the thresholds for rainbow…
We define a generalization of threshold graphs which we call $k$-rainbow threshold graphs. We show that the collection of $k$-rainbow threshold graphs do not satisfy the $0$-$1$ law for first order logic and that asymptotically almost…
In this contribution I present results achieved recently in the field of the OT forecast that push further the limit of the accuracy of the OT forecasts and open to new perspectives in this field.
We survey recent developments on the Restriction conjecture.
We report on the first experimental demonstration of the broadband "trapped rainbow" in the visible frequency range using an adiabatically tapered waveguide. Being a distinct case of the slow light phenomenon, the trapped rainbow effect…
We discuss recent progress many problems in random matrix theory of a combinatorial nature, including several breakthroughs that solve long standing famous conjectures.
The concept of rainbow connection was introduced by Chartrand et al. in 2008. It is fairly interesting and recently quite a lot papers have been published about it. In this survey we attempt to bring together most of the results and papers…
We report the first experimental observation of trapped rainbow1 in graded metallic gratings2-4, designed to validate theoretical predictions for this new class of plasmonic structures. One-dimensional tapered gratings were fabricated and…
Let $G$ be an edge-colored graph. The color degree of a vertex $v$ of $G$, is defined as the number of colors of the edges incident to $v$. The color number of $G$ is defined as the number of colors of the edges in $G$. A rainbow triangle…
We study the limit theory of large threshold graphs and apply this to a variety of models for random threshold graphs. The results give a nice set of examples for the emerging theory of graph limits.
We call an edge colouring of a graph G a rainbow colouring if every pair of vertices is joined by a rainbow path, i.e., a path where no two edges have the same colour. The minimum number of colours required for a rainbow colouring of the…
A conjecture of the first two authors is that $n$ matchings of size $n$ in any graph have a rainbow matching of size $n-1$. We prove a lower bound of $\frac{2}{3}n-1$, improving on the trivial $\frac{1}{2}n$, and an analogous result for…
A rainbow stacking of $r$-edge-colorings $\chi_1, \ldots, \chi_m$ of the complete graph on $n$ vertices is a way of superimposing $\chi_1, \ldots, \chi_m$ so that no edges of the same color are superimposed on each other. We determine a…
We review some recent development in the theory of spatial extremes related to Pareto Processes and modeling of threshold exceedances. We provide theoretical background, methodology for modeling, simulation and inference as well as an…
We seek conditions under which colorings of various vector spaces are guaranteed to have a copy of a unit equilateral triangle, having each vertex in a different color class. In particular, we explore the analogous question in the setting…
We show that the threshold for having a rainbow copy of a power of a Hamilton cycle in a randomly edge colored copy of $G_{n,p}$ is within a constant factor of the uncolored threshold. Our proof requires $(1+\varepsilon)$ times the minimum…
Let $G_{n,p}^{[\kappa]}$ denote the space of $n$-vertex edge coloured graphs, where each edge occurs independently with probability $p$. The colour of each existing edge is chosen independently and uniformly at random from the set…
A brief overview of the status of color transparency experiments is presented. We report on the first complete calculations of color transparency within a perturbative QCD framework. We also comment on the underlying factorization method…
In this paper, we introduce the notion of the rainbow neighbourhood and a related graph parameter namely, the rainbow neighbourhood number of a graph $G$. We report on preliminary results thereof. We also establish a necessary and…
A rainbow matching for (not necessarily distinct) sets F_1,...,F_k of hypergraph edges is a matching consisting of k edges, one from each F_i. The aim of the paper is twofold - to put order in the multitude of conjectures that relate to…