Proper rainbow connection number of graphs

A path in an edge-coloured graph is called \emph{rainbow path} if its edges receive pairwise distinct colours. An said to be connected} any two vertices of the are connected by a rainbow path. The minimum $k$ for which there exists such edge-colouring connection number $rc(G)$ $G.$ Recently, Bau et al. \cite{BJJKM2018} introduced this concept with additional requirement that must proper. %An proper \emph{properly \emph{proper number} $G$, denoted $prc(G)$, colours needed order make it properly connected. In paper we first prove improved upper bound $prc(G) \leq n$ every $G$ $n \geq 3.$ Next show difference - rc(G)$ can arbitrarily large. Finally, present several sufficient conditions classes satisfying = \chi'(G).$