A degree sum condition for graphs to be covered by two cycles

Let GG be a kk-connected graph of order nn. In [1], Bondy (1980) considered a degree sum condition for a graph to have a Hamiltonian cycle, say, to be covered by one cycle. He proved that if σk+1(G)>(k+1)(n−1)/2σk+1(G)>(k+1)(n−1)/2, then GG has a Hamiltonian cycle. On the other hand, concerning a degree sum condition for a graph to be covered by two cycles, Enomoto et al. (1995) [4] proved that if k=1k=1 and σ3(G)≥nσ3(G)≥n, then GG can be covered by two cycles. By these results, we conjecture that if σ2k+1(G)>(2k+1)(n−1)/3σ2k+1(G)>(2k+1)(n−1)/3, then GG can be covered by two cycles. In this paper, we prove the case k=2k=2 of this conjecture. In fact, we prove a stronger result; if GG is 2-connected with σ5(G)≥5(n−1)/3σ5(G)≥5(n−1)/3, then GG can be covered by two cycles, or GG belongs to an exceptional class.