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.

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On the Rainbow Vertex-Connection

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 �-connected graph G and an integer k with 1 � k � �, the rainbow kconnectivity rck(G) of G is defined as the minimum integer j for which there exists a j-edge-coloring of G such that any two distinct vertices of G are connected by k internally disjoint rainbow paths. Denote by Kr,r an r-regular complete bipartite graph. Chartrand et al. in “G. Chartrand, G.L. Johns, K.A. McKeon, P. Zhang, The rainbow connectivity of a graph, Networks 54(2009), 75-81” left an open question of determining an integer g(k) for which the rainbow k-connectivity of Kr,r is

Note on the Rainbow k-Connectivity of Regular Complete Bipartite Graphs

A path in an edge-coloring graph G, where adjacent edges may be colored the same, is called a rainbow path if no two edges of G are colored the same. A nontrivial connected graph G is rainbow connected if for any two vertices of G there is a rainbow path connecting them. The rainbow connection number of G, denoted rc(G), is defined as the minimum number of colors by using which there is coloring such that G is rainbow connected. In this paper, we study the rainbow connection numbers of line graphs of triangle-free graphs, and particularly, of 2-connected triangle-free graphs according to their ear decompositions.

Rainbow connection numbers of line graphs.

A path in an edge-colored graph, where adjacent edges may be colored the same, is a rainbow path if no two edges of it are colored the same. For any two vertices u and v of G, a rainbow ui v geodesic in G is a rainbow ui v path of length d(u; v), where d(u; v) is the distance between u and v. The graph G is strongly rainbow connected if there exists a rainbow ui v geodesic for any two vertices u and v in G. The strong rainbow connection number of G, denoted by src(G), is the minimum number of colors that are needed in order to make G strongly rainbow connected. In this paper, we first give a sharp upper bound for src(G) in terms of the number of edge-disjoint triangles in a graph G, and give a necessary and sufficient condition for the equality. We next investigate the graphs with large strong rainbow connection numbers. Chartrand et al. obtained that src(G) = m if and only if G is a tree, we will show that src(G) 6 mi 1, and characterize the graphs G with src(G) = mi 2 where m is the number of edges of G.

/pdf/on-the-strong-rainbow-connection-of-a-graph-1ykhs1vfgh.pdf

On the Strong Rainbow Connection of a Graph

The generalized k-connectivity κk(G) of a graph G was introduced by Chartrand et al. in 1984. As a natural counterpart of this concept, Li et al. in 2011 introduced the concept of generalized k-edge-connectivity which is defined as λk(G) = min{λ(S): S ⊆ V (G) and |S| = k}, where λ(S) denotes the maximum number l of pairwise edge-disjoint trees T1, T2, …, Tl in G such that S ⊆ V (Ti) for 1 ⩽ i ⩽ l. In this paper we prove that for any two connected graphs G and H we have λ3(G □ H) ⩾ λ3(G) + λ3(H), where G □ H is the Cartesian product of G and H. Moreover, the bound is sharp. We also obtain the precise values for the generalized 3-edge-connectivity of the Cartesian product of some special graph classes.

/pdf/generalized-3-edge-connectivity-of-cartesian-product-graphs-49gcdpctqg.pdf

Generalized 3-edge-connectivity of Cartesian product graphs

Generalized connectivity introduced by Hager (1985) has been studied extensively in undirected graphs and become an established area in undirected graph theory. For connectivity problems, directed graphs can be considered as generalizations of undirected graphs. In this paper, we introduce a natural extension of generalized $k$-connectivity of undirected graphs to directed graphs (we call it strong subgraph $k$-connectivity) by replacing connectivity with strong connectivity. We prove NP-completeness results and the existence of polynomial algorithms. We show that strong subgraph $k$--connectivity is, in a sense, harder to compute than generalized $k$-connectivity. However, strong subgraph $k$-connectivity can be computed in polynomial time for semicomplete digraphs and symmetric digraphs. We also provide sharp bounds on strong subgraph $k$-connectivity and pose some open questions.

Yuefang Sun

Papers

Note on the Rainbow k-Connectivity of Regular Complete Bipartite Graphs

Rainbow connection numbers of line graphs.

On the Strong Rainbow Connection of a Graph

Generalized 3-edge-connectivity of Cartesian product graphs

Strong subgraph k-connectivity