On the minimum degree and the proper connection number of graphs

An edge-coloured graph G is called properly connected if any two vertices are connected by a path whose edges are properly coloured. The proper connection number of a graph G, denoted by pc(G), is the smallest number of colours that are needed in order to make G properly connected. In this paper we consider sufficient conditions in terms of the ratio between minimum degree and order of a 2-connected graph G implying that G has proper connection number 2.