Edge degree conditions for subpancyclicity in line graphs

In this paper, two best possible edge degree conditions are given for the line graph L(G) of a graph G with girth at least 4 or 5 to be subpancyclic, i.e., L(G) contains a cycle of length k, for each k between 3 and the circumference of L(G). In [5] the following conjecture is made: If G is a graph such that the degree sum of any pair of adjacent vertices in G is greater than (√8n + 1 + 1)/2, then the line graph L(G) of G is pancyclic whenever L(G) is Hamiltonian, unless G is isomorphic to C₄, C₅, or the Petersen graph. Our results show that the conjecture is true for those graphs of order n≥72 with girth at least 4.