Rainbow Connection In Sparse Graphs

An edge-coloured connected graph $G = (V,E)$ is called rainbow-connected if each pair of distinct vertices of $G$ is connected by a path whose edges have distinct colours. The rainbow connection number of $G$, denoted by $ \text{rc}(G)$, is the minimum number of colours such that $G$ is rainbow-connected. In this paper we prove that $ \text{rc}(G) \le k $ if $ |V (G)| = n $ and \( |E(G)| \ge \binom{n-k+1}{2} + k -1 \) for all integers $n$ and $k$ with $n − 6 \le k \le n − 3 $. We also show that this bound is tight.