An edge-coloured connected graph is rainbow connected if each two vertices are connected by a path whose edges have distinct colours. If such a colouring uses colours then is called -rainbow connected. The rainbow connection number of , denoted by , is the minimum such that is -rainbow connected. Even the problem to decide whether is NP-complete. It has been shown that if is a connected graph of order and size with , then .
In this paper we will present sufficient conditions for graphs of this size to fulfil depending on vertex degrees.