Heavy cycles in hamiltonian weighted graphs

Let G be a 2-connected weighted graph such that the minimum weighted degree is at least d. In [1], Bondy and Fan proved that either G contains a cycle of weight at least 2d or every heaviest cycle in G is a hamiltonian cycle. If G is not hamiltonian, this theorem implies the existence of a cycle of weight at least 2d, but in case of G is hamiltonian we cannot decide whether G has a heavy cycle or not. In this paper, we prove that if G is triangle-free, then G has a cycle of weight at least 2d even in case of G is hamiltonian.