A Rainbow Connectivity Threshold for Random Graph Families

  title={A Rainbow Connectivity Threshold for Random Graph Families},
  author={Peter Bradshaw and Bojan Mohar},
Abstract. Given a family G of graphs on a common vertex set X, we say that G is rainbow connected if for every vertex pair u, v ∈ X, there exists a path from u to v that uses at most one edge from each graph in G. We consider the case that G contains s graphs, each sampled randomly from G(n, p), with n = |X| and p = c logn sn , where c > 1 is a constant. We show that when s is sufficiently large, G is a.a.s. rainbow connected, and when s is sufficiently small, G is a.a.s. not rainbow connected… 



The Diameter of Sparse Random Graphs

The diameter of a random graph G(n,p) for various ranges of p close to the phase transition point for connectivity is considered using the convention that the diameter of G is the maximum diameter of its connected components.

Diameters of Random Graphs

The present paper deals with the family G approximate (n,E) of all labeled graphs that have n nodes and E edges, which is a combinatorial foundation for investigations of the average-case behavior of various graph-theoretic algorithms.

Rainbow Hamilton cycles in random graphs and hypergraphs

A general tool for tackling problems related to finding “nicely edge-colored” structures in random graphs/hypergraphs is provided, based on an ingenious coupling idea of McDiarmid.

On a rainbow version of Dirac's theorem

For a collection G={G1,⋯,Gs} of not necessarily distinct graphs on the same vertex set V , a graph H with vertices in V is a G ‐transversal if there exists a bijection ϕ:E(H)→[s] such that e∈E(Gϕ(e))

Some Remarks on Rainbow Connectivity

This article proposes a very simple approach to studying rainbow connectivity in graphs and gives a unified proof of several known results, as well as some new ones.

Random subgraphs of properly edge-coloured complete graphs and long rainbow cycles

A subgraph of an edge-coloured complete graph is called rainbow if all its edges have different colours. In 1980 Hahn conjectured that every properly edge-coloured complete graph Kn has a rainbow

A rainbow version of Mantel's Theorem

Mantel's Theorem from 1907 is one of the oldest results in graph theory: every simple $n$-vertex graph with more than $\frac{1}{4}n^2$ edges contains a triangle. The theorem has been generalized in

Rainbow connection in graphs

Let $G$ be a nontrivial connected graph on which is defined a coloring $c\: E(G) \rightarrow \lbrace 1, 2, \ldots , k\rbrace $, $k \in {\mathbb{N}}$, of the edges of $G$, where adjacent edges may be

Transversals and Bipancyclicity in Bipartite Graph Families

A bipartite graph is called bipancyclic if it contains cycles of every even length from four up to the number of vertices in the graph. A theorem of Schmeichel and Mitchem states that for $n

The diameter of sparse random graphs

An expression of the form c ln n + o(ln n) for the diameter of a sparse random graph with a specified degree sequence is derived, and is applicable to the classical random graph Gn,p with np = Θ(1) + 1, as well as certain random power law graphs.