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

Given an integer k>=1 and any graph G, the path graph P"k(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-1 in G, and their union forms either a cycle or a path of length k+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, P"1(G) is the line graph of G. For k=1,2 results on connectivity of P"k(G) have been given for several authors. In this work, we present a sufficient condition to guarantee that P"k(G) is connected for k>=2 if the girth of G is at least (k+3)/2 and its minimum degree is at least 4. Furthermore, we determine a lower bound of the vertex-connectivity of P"k(G) if the girth is at least k+1 and the minimum degree is at least r+1 where r>=2 is an integer.