A sufficient condition for PkPk-path graphs being r-connected

Given an integer k⩾1k⩾1 and any graph G,G, the path graph Pk(G)Pk(G) has for vertices the paths of length k in G  , and two vertices are joined by an edge if and only if the intersection of the corresponding paths forms a path of length k-1k-1 in G  , and their union forms either a cycle or a path of length k+1k+1.Path graphs were investigated by Broersma and Hoede [Path graphs, J. Graph Theory 13 (1989), 427–444] as a natural generalization of line graphs. In fact, P1(G)P1(G) is the line graph of GG. For k=1,2k=1,2 results on connectivity of Pk(G)Pk(G) have been given for several authors. In this work, we present a sufficient condition to guarantee that Pk(G)Pk(G) is connected for k⩾2k⩾2 if the girth of G   is at least (k+3)/2(k+3)/2 and its minimum degree is at least 4. Furthermore, we determine a lower bound of the vertex-connectivity of Pk(G)Pk(G) if the girth is at least k+1k+1 and the minimum degree is at least r+1r+1 where r⩾2r⩾2 is an integer.