Some Bounds on the Rainbow Connection Number of 3-, 4- and 5-connected Graphs

The rainbow connection number, $rc(G)$, of a connected graph $G$ is the minimum number of colors needed to color its edges so that every pair of vertices is connected by at least one path in which no two edges are colored the same. We show that for $\kappa =3$ or $\kappa = 4$, every $\kappa$-connected graph $G$ on $n$ vertices with diameter $\frac{n}{\kappa}-c$ satisfies $rc(G) \leq \frac{n}{\kappa} + 15c + 18$. We also show that for every maximal planar graph $G$, $rc(G) \leq \frac{n}{\kappa} + 36$. This proves a conjecture of Li et al. for graphs with large diameter and maximal planar graphs.