On circuits and pancyclic line graphs

Clark proved that L(G) is hamiltonian if G is a connected graph of order n 2 6 such that deg u + deg v 2 n 1 p(n) for every edge uv of G, where p(n ) = 0 if n is even and p(n) = 1 if n is odd. Here it is shown that the bound n 1 dn) can be decreased to (2n + 1)/3 if every bridge of G is incident with a vertex of degree 1, which is a necessary condition for hamiltonicity of L(G). Moreover, the conclusion that L(G) is hamiltonian can be strengthened to the conclusion that L ( G ) is pancyclic. Lesniak-Foster and Williamson proved that G contains a spanning closed trail if IV(G)J = n 2 6, 6 ( G ) 2 2 and deg u + deg v 2 n 1 for every pair of nonadjacent vertices u and v. The bound n 1 can be decreased to (2n + 3)/3 if G is connected and bridgeless, which is necessary for G to have a spanning closed trail.