Rainbow connectivity of randomly perturbed graphs

On Rainbow-k-Connectivity of Random Graphs

A path in an edge-colored graph is called a rainbow path if the edges on it have distinct colors. For k ≥ 1, the rainbow-k-connectivity of a graph G, denoted rck(G), is the minimum number of colors required to color the edges of G in such a way that every two distinct vertices are connected by at least k internally vertex-disjoint rainbow paths. In this paper, we study rainbow-k-connectivity in...

Rainbow Connectivity of Sparse Random Graphs

An edge colored graph G is rainbow edge connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connectivity of a connected graph G, denoted by rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this work we study the rainbow connectivity of binomial random graphs at the connectivity threshold p = logn+ω n...

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...