A sufficient condition for pancyclic graphs

In 2005, Rahman and Kaykobad proved that if G is a connected graph of order n such that d(x)+d(y)+d(x,y)⩾n+1 for each pair x, y of distinct nonadjacent vertices in G, where d(x,y) is the length of a shortest path between x and y in G, then G has a Hamiltonian path [Inform. Process. Lett. 94 (2005) 37–41]. In 2006 Li proved that if G is a 2-connected graph of order n⩾3 such that d(x)+d(y)+d(x,y)⩾n+2 for each pair x,y of nonadjacent vertices in G, then G is pancyclic or G=Kn/2,n/2 where n⩾4 is an even integer [Inform. Process. Lett. 98 (2006) 159–161]. In this work we prove that if G is a 2-connected graph of order n such that d(x)+d(y)+d(x,y)⩾n+1 for all pairs x, y of distinct nonadjacent vertices in G, then G is pancyclic or G belongs to one of four specified families of graphs.