Tilings in Randomly Perturbed Dense Graphs

Bounded-Degree Spanning Trees in Randomly Perturbed Graphs

We show that for any xed dense graph G and bounded-degree tree T on the same number of vertices, a modest random perturbation of G will typically contain a copy of T . This combines the viewpoints of the well-studied problems of embedding trees into xed dense graphs and into random graphs, and extends a sizeable body of existing research on randomly perturbed graphs. Speci cally, we show that t...

Expansion and Lack Thereof in Randomly Perturbed Graphs

Expansion in randomly perturbed connected graphs and its consequences

We propose a unified approach, based on expansion, to analyzing combinatorial properties of random perturbations of connected graphs. Given an n-vertex connected graph G, form a random supergraph G∗ of G by turning every pair of vertices of G into an edge with probability ε n , where ε is a small positive constant. This perturbation model has been studied previously in several contexts, includi...

Error distribution in randomly perturbed orbits.

Given an observable f defined on the phase space of some dynamical system generated by the mapT, we consider the error between the value of the function f(Tnx0) computed at time n along the orbit with initial condition x0, and the value f(Tn x0) of the same observable computed by replacing the map Tn with the composition of maps T(omega)(n)o...oT(omega)1, where each T(omega) is chosen randomly,...

Tilings by Regular Polygons Iii: Dodecagon-dense Tilings

In Tilings and Patterns, Grünbaum and Shephard claim that there are only four kuniform tilings by regular polygons (for some k) that have a dodecagon incident at every vertex. In fact, there are many others. We show that the tilings that satisfy this requirement are either the uniform 4.6.12 tiling, or else fall into one of two infinite classes of such tilings. One of these infinite classes can...