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

The generalized k-connectivity $$\kappa _{k}(G)$$
 and the generalized k-edge-connectivity $$\lambda _k(G)$$
 of a graph G, also known as the tree connectivities, were introduced by Hager (J Comb Theory Ser B 38:179–189, 1985) and Li et al. (Discret Math Theor Comput Sci 14:43–54, 2012), respectively. In this paper, we study these invariants for Cayley graphs on Abelian groups with degree 3 or 4. When G is cubic, we prove $$\kappa _k(G) = \lambda _k(G) = 2$$
 for $$3\le k\le 6$$
 and $$\kappa _k(G) = \lambda _k(G) = 1$$
 for $$7\le k\le n$$
 . When G has degree 4, we obtain $$\kappa _{3}(G)=\lambda _3(G)=3$$
 , $$\lambda _k(G) = 2$$
 and $$\kappa _k(G)\le 2$$
 for $$k\ge 8$$
 , and $$\kappa _k(G) = 2$$
 for $$k=n-1,n$$
 .

Tree Connectivities of Cayley Graphs on Abelian Groups with Small Degrees

Rainbow matchings in edge-colored complete split graphs

The concept of k-connectivity $$\kappa '_{k}(G)$$źkź(G) of a graph G, introduced by Chartrand in 1984, is a generalization of the cut-version of the classical connectivity. Another generalized connectivity of a graph G, named the generalized k-connectivity $$\kappa _{k}(G)$$źk(G), mentioned by Hager in 1985, is a natural generalization of the path-version of the classical connectivity. In this paper, we get the lower and upper bounds for the difference of these two parameters by showing that for a connected graph G of order n, if $$\kappa '_k(G)\ne n-k+1$$źkź(G)źn-k+1 where $$k\ge 3$$kź3, then $$0\le \kappa '_k(G)-\kappa _k(G)\le n-k-1$$0≤źkź(G)-źk(G)≤n-k-1; otherwise, $$-\lfloor \frac{k}{2}\rfloor +1\le \kappa '_k(G)-\kappa _k(G)\le n-k$$-źk2ź+1≤źkź(G)-źk(G)≤n-k. Moreover, all of these bounds are sharp. Some specific study is focused for the case $$k=3$$k=3. As results, we characterize the graphs with $$\kappa '_3(G)=\kappa _3(G)=t$$ź3ź(G)=ź3(G)=t for $$t\in \{1, n-3, n-2\}$$tź{1,n-3,n-2}, and give a necessary condition for $$\kappa '_3(G)=\kappa _3(G)$$ź3ź(G)=ź3(G) by showing that for a connected graph G of order n and size m, if $$\kappa '_3(G)=\kappa _3(G)=t$$ź3ź(G)=ź3(G)=t where $$1\le t\le n-3$$1≤t≤n-3, then $$m\le {n-2\atopwithdelims ()2}+2t$$m≤n-22+2t. Moreover, the unique extremal graph is given for the equality to hold.

On the difference of two generalized connectivities of a graph

An edge-colored graph G, where adjacent edges may have the same color, is rainbow connected if every two vertices of G are connected by a path whose edges have distinct colors. A graph G is d-rainbow connected if one can use d colors to make G rainbow connected. For integers n and d let t(n, d) denote the minimum size (number of edges) in d-rainbow connected graphs of order n. Schiermeyer got some exact values and upper bounds for t(n, d). However, he did not present a lower bound of t(n, d) for $${3 \leq d < \lceil\frac{n}{2}\rceil}$$ . In this paper, we improve his lower bound of t(n, 2), and get a lower bound of t(n, d) for $${3 \leq d < \lceil\frac{n}{2}\rceil}$$ .

Note on Minimally d-Rainbow Connected Graphs

A path in an edge-colored graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of the path are colored the same. For a κ-connected graph G and an integer k with 1 ≤ k ≤ κ, the rainbow k-connectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that every two distinct vertices of G are connected by k internally disjoint rainbow paths. This paper is to investigate the rainbow k-connectivity of complete graphs. We improve the upper bound of f(k) from (k + 1) by Chartrand et al. to ck 3 2 + C(k), where c is a constant, C(k) = o(k 3 2 ), and f(k) is the integer such that if n ≥ f(k) then rck(Kn) = 2. Recently, we saw that Dellamonica et al. got the best possible upper bound 2k, which is linear in k. However, our proof is more structural or constructive.

/pdf/on-the-rainbow-k-connectivity-of-complete-graphs-1cswb4kyve.pdf

Yuefang Sun

Papers

Tree Connectivities of Cayley Graphs on Abelian Groups with Small Degrees

Rainbow matchings in edge-colored complete split graphs

On the difference of two generalized connectivities of a graph

Note on Minimally d-Rainbow Connected Graphs

On the rainbow k-connectivity of complete graphs.