Path-comprehensive and vertex-pancyclic properties of super line graph L2(G)L2(G)

The super line graph of index rr of a graph GG, denoted by Lr(G)Lr(G), was introduced by Bagga, Beineke and Varma. They showed that if GG is connected and has at least two edges, then L2(G)L2(G) is pancyclic. An improved result was also obtained: if GG has no isolated edges, then L2(G)L2(G) is vertex-pancyclic. A graph GG of order nn is path-comprehensive   if every pair of vertices are joined by paths of all lengths 2,3,…,n−12,3,…,n−1. In this paper, we obtain that if GG has no isolated edges, then L2(G)L2(G) is path-comprehensive, and that if GG has at most one isolated edge, then L2(G)L2(G) is vertex-pancyclic, which gives a positive answer to the question due to Bagga et al. whether L2(G)L2(G) is also vertex-pancyclic even if one isolated edge is permitted.