We will review some of the major results in random graphs and some of the more challenging open problems. We will cover algorithmic and structural questions. We will touch on newer models, including those related to the WWW.

Random graphs

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 that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow $k$-connectivity, $k$-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems or questions.

Rainbow connections of graphs -- A survey

The concept of rainbow connection was introduced by Chartrand et al. [14] in 2008. It is 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 that dealt with it. We begin with an introduction, and then try to organize the work into five categories, including (strong) rainbow connection number, rainbow k-connectivity, k-rainbow index, rainbow vertex-connection number, algorithms and computational complexity. This survey also contains some conjectures, open problems and questions.

Rainbow Connections of Graphs: A Survey

It is shown that, for each fixed $k$, the problem of finding $k$ pairwise vertex-disjoint directed paths between given pairs of terminals in a directed planar graph is solvable in polynomial time.

Finding k disjoint paths in a directed planar graph

A vertex-colored graph is rainbow vertex-connected if any two vertices are connected by a path whose internal vertices have distinct colors. The rainbow vertex-connection of a connected graph G, denoted by rvc(G), is the smallest number of colors that are needed in order to make G rainbow vertexconnected. It was proved that if G is a graph of order n with minimum degree δ, then rvc(G) < 11n/δ. In this paper, we show that rvc(G) ≤ 3n/(δ+1)+5 for δ ≥ √ n − 1 − 1 and n ≥ 290, while rvc(G) ≤ 4n/(δ + 1) + 5 for 16 ≤ δ ≤ √ n − 1 − 2 and rvc(G) ≤ 4n/(δ + 1) + C(δ) for 6 ≤ δ ≤ 15, where C(δ) = e 3 log(�3+2�2+3) 3(log 3 1) � 3 − 2. We also prove that rvc(G) ≤ 3n/4 − 2 for δ = 3, rvc(G) ≤ 3n/5 − 8/5 for δ = 4 and rvc(G) ≤ n/2 − 2 for δ = 5. Moreover, an example constructed by Caro et al. shows that when δ ≥ √ n − 1−1 and δ = 3,4,5, our bounds are seen to be tight up to additive constants.

/pdf/on-the-rainbow-vertex-connection-2o8m8seac6.pdf

On the Rainbow Vertex-Connection

1. Introduction (Motivation and definitions, Terminology and notations).- 2. (Strong) Rainbow connection number(Basic results, Upper bounds for rainbow connection number, For some graph classes, For dense and sparse graphs, For graph operations, An upper bound for strong rainbow connection number).- 3. Rainbow k-connectivity.- 4. k-rainbow index.- 5. Rainbow vertex-connection number.- 6. Algorithms and computational complexity.- References.

Rainbow Connections of Graphs

The concept of rainbow connection was introduced by Chartrand, Johns, McKeon and Zhang in 2008. Nowadays it has become a new and active subject in graph theory. There is a book on this topic by Li and Sun in 2012, and a survey paper by Li, Shi and Sun in 2013. More and more researchers are working in this field, and many new papers have been published in journals. In this survey we attempt to bring together most of the new results and papers that deal with this topic. We begin with an introduction, and then try to organize the work into the following categories, rainbow connection coloring of edge-version, rainbow connection coloring of vertex-version, rainbow k-connectivity, rainbow index, rainbow connection coloring of total-version, rainbow connection on digraphs, rainbow connection on hypergraphs. This survey also contains some conjectures, open problems and questions for further study.

/pdf/an-updated-survey-on-rainbow-connections-of-graphs-a-dynamic-1opanmlual.pdf

An updated survey on rainbow connections of graphs - a dynamic survey

The generalized $k$-connectivity $\kappa_k(G)$ of a graph $G$ was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized $k$-edge-connectivity $\lambda_k(G)$. For general $k$, the generalized $k$-edge-connectivity of a complete graph is obtained. For $k\geq 3$, tight upper and lower bounds of $\kappa_k(G)$ and $\lambda_k(G)$ are given for a connected graph $G$ of order $n$, that is, $1\leq \kappa_k(G)\leq n-\lceil\frac{k}{2}\rceil$ and $1\leq \lambda_k(G)\leq n-\lceil\frac{k}{2}\rceil$. Graphs of order $n$ such that $\kappa_k(G)=n-\lceil\frac{k}{2}\rceil$ and $\lambda_k(G)=n-\lceil\frac{k}{2}\rceil$ are characterized, respectively. Nordhaus-Gaddum-type results for the generalized $k$-connectivity are also obtained. For $k=3$, we study the relation between the edge-connectivity and the generalized 3-edge-connectivity of a graph. Upper and lower bounds of $\lambda_3(G)$ for a graph $G$ in terms of the edge-connectivity $\lambda$ of $G$ are obtained, that is, $\frac{3\lambda-2}{4}\leq \lambda_3(G)\leq \lambda$, and two graph classes are given showing that the upper and lower bounds are tight. From these bounds, we obtain that $\lambda(G)-1\leq \lambda_3(G)\leq \lambda(G)$ if $G$ is a connected planar graph, and the relation between the generalized 3-connectivity and generalized 3-edge-connectivity of a graph and its line graph.

Yuefang Sun

Papers

Rainbow connections of graphs -- A survey

Rainbow Connections of Graphs

Rainbow Connections of Graphs: A Survey

An updated survey on rainbow connections of graphs - a dynamic survey

On the generalized (edge-)connectivity of graphs