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

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.

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On the Rainbow Vertex-Connection

The rainbow connection number, rc(G), of a connected graph G is the minimum number of colours needed to colour its edges, so that every pair of its vertices is connected by at least one path in which no two edges are coloured the same. In this note we show that for every bridgeless graph G with radius r, rc(G) <= r(r + 2). We demonstrate that this bound is the best possible for rc(G) as a function of r, not just for bridgeless graphs, but also for graphs of any stronger connectivity. It may be noted that for a general 1-connected graph G, rc(G) can be arbitrarily larger than its radius (Star graph for instance). We further show that for every bridgeless graph G with radius r and chordality (size of a largest induced cycle) k, rc(G) <= rk. 
It is known that computing rc(G) is NP-Hard [Chakraborty et al., 2009]. Here, we present a (r+3)-factor approximation algorithm which runs in O(nm) time and a (d+3)-factor approximation algorithm which runs in O(dm) time to rainbow colour any connected graph G on n vertices, with m edges, diameter d and radius r.

Rainbow Connection Number and Radius

An edge-colored graph $G$ is rainbow connected if any two vertices are connected by a path whose edges have distinct colors. The rainbow connection number 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 paper, we proved that $rc(G)\leq 3(n+1)/5$ for all $3$-connected graphs.

Rainbow connection in $3$-connected graphs

The rainbow connection number of a connected graph is the minimum number of colors needed to color its edges, so that every pair of its vertices is connected by at least one path in which no two edges are colored the same. In this article we show that for every connected graph on n vertices with minimum degree δ, the rainbow connection number is upper bounded by 3n/(δ + 1) + 3. This solves an open problem from Schiermeyer (Combinatorial Algorithms, Springer, Berlin/Hiedelberg, 2009, pp. 432–437), improving the previously best known bound of 20n/δ (J Graph Theory 63 (2010), 185–191). This bound is tight up to additive factors by a construction mentioned in Caro et al. (Electr J Combin 15(R57) (2008), 1).

As an intermediate step we obtain an upper bound of 3n/(δ + 1) − 2 on the size of a connected two-step dominating set in a connected graph of order n and minimum degree δ. This bound is tight up to an additive constant of 2. This result may be of independent interest.

We also show that for every connected graph G with minimum degree at least 2, the rainbow connection number, rc(G), is upper bounded by Γc(G) + 2, where Γc(G) is the connected domination number of G. Bounds of the form diameter(G)⩽rc(G)⩽diameter(G) + c, 1⩽c⩽4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, asteroidal triple-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G)⩽3·radius(G). In most of these cases, we also demonstrate the tightness of the bounds. © 2012 Wiley Periodicals, Inc.

Rainbow connection number and connected dominating sets

Cubicity, boxicity, and vertex cover

Rainbow Connection Number and Connected Dominating Sets

Rainbow connection number rc(G) of a connected graph G is the minimum number of colours needed to colour the edges of G, so that every pair of vertices is connected by at least one path in which no two edges are coloured the same. In this paper we show that for every connected graph G, with minimum degree at least 2, the rainbow connection number is upper bounded by {\gamma}_c(G) + 2, where {\gamma}_c(G) is the connected domination number of G. Bounds of the form diameter(G) \leq rc(G) \leq diameter(G) + c, 1 \leq c \leq 4, for many special graph classes follow as easy corollaries from this result. This includes interval graphs, AT-free graphs, circular arc graphs, threshold graphs, and chain graphs all with minimum degree at least 2 and connected. We also show that every bridge-less chordal graph G has rc(G) \leq 3.radius(G). In most of these cases, we also demonstrate the tightness of the bounds. An extension of this idea to two-step dominating sets is used to show that for every connected graph on n vertices with minimum degree {\delta}, the rainbow connection number is upper bounded by 3n/({\delta} + 1) + 3. This solves an open problem of Schiermeyer (2009), improving the previously best known bound of 20n/{\delta} by Krivelevich and Yuster (2010). Moreover, this bound is seen to be tight up to additive factors by a construction of Caro et al. (2008).

Anita Das

Papers

Rainbow connection number and connected dominating sets

Cubicity, boxicity, and vertex cover

Rainbow Connection Number and Connected Dominating Sets

Rainbow Connection Number and Connected Dominating Sets

Tree 3-spanners in 2-sep chordal graphs: Characterization and algorithms