Proper connection number 2, connectivity, and forbidden subgraphs

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 connectivity and forbidden subgraphs, implying a graph to have proper connection number 2.