Let $G$ be a nontrivial connected graph on which is defined a coloring $cE\left(G\right)\to \{1,2,...,k\}$, $k\in \mathbb{N}$, of the edges of $G$, where adjacent edges may be colored the same. A path $P$ in $G$ is a rainbow path if no two edges of $P$ are colored the same. The graph $G$ is rainbow-connected if $G$ contains a rainbow $u-v$ path for every two vertices $u$ and $v$ of $G$. The minimum $k$ for which there exists such a $k$-edge coloring is the rainbow connection number $\mathrm{r}c\left(G\right)$ of $G$. If for every pair $u,v$ of distinct vertices, $G$ contains a rainbow $u-v$ geodesic, then $G$ is...

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 rc(G), is the smallest number of colors that are needed in order to make G rainbow connected. In this paper we show that rc(G) ≤ 3 if |E(G)| ≥ [...] + 2, and rc(G) ≤ 4 if |E(G)| ≥ [...] + 3. These bounds are sharp.

A path in an edge-colored graph G is rainbow if no two edges of the path are colored the same. The rainbow connection number rc(G) of G is the smallest integer k for which there exists a k-edge-coloring of G such that every pair of distinct vertices of G is connected by a rainbow path. Let f(d) denote the minimum number such that rc(G) ≤ f(d) for each bridgeless graph G with diameter d. In this paper, we shall show that 7 ≤ f(3) ≤ 9.

An arc-coloured digraph D = (V,A) is said to be rainbow connected if for every pair {u, v} ⊆ V there is a directed uv-path all whose arcs have different colours and a directed vu-path all whose arcs have different colours. The minimum number of colours required to make the digraph D rainbow connected is called the rainbow connection number of D, denoted rc⃗ (D). A cactus is a digraph where each arc belongs to exactly one directed cycle. In this paper we give sharp upper and lower bounds for the...