Edge-connectivity in image-path graphs

The Pk(G)Pk(G)-path graph corresponding to a graph G has for vertices the set of all paths of length k in G. 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 introduced by Broersma and Hoede (J. Graph. Theory 13 (1989) 427) as a generalization of line graphs, because for k=1k=1, path graphs are just line graphs. Results on the edge-connectivity of line graphs are given by Chartrand and Stewart (Math. Ann. 182 (1969) 170), later by Zamfirescu (Math. Ann. 187 (1970) 305), and by Jixiang Meng (Graph Theory Notes of New York XL (2001) 12). The connectivity of PkPk-path graphs has been studied by Knor and Niepel (Graph Theory 20 (2000) 181), where they proved a necessary and sufficient condition for the Pk(G)Pk(G)-path graphs to be disconnected, assuming that G has girth of at least k+1k+1. Going one step further, we prove in this work that the edge-connectivity of Pk(G)Pk(G) is at least λ(Pk(G))⩾δ(G)-1λ(Pk(G))⩾δ(G)-1 for a graph G of girth at least k+1k+1 and minimum degree δ(G)⩾2δ(G)⩾2. Furthermore, we show λ(Pk(G))⩾2δ(G)-2λ(Pk(G))⩾2δ(G)-2 provided that δ(G)⩾3δ(G)⩾3.