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

Let G be a k-connected graph of order n. In [1], Bondy 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)/2, then G 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. [4] proved that if k = 1 and σ3(G) ≥ n, then G can be covered by two cycles. By these results, we conjecture that if σ2k+1(G) > (2k + 1)(n − 1)/3, then G can be covered by two cycles. In this paper, we prove the case k = 2 of this conjecture. In fact, we prove stronger result; if G is 2-connected with σ5(G) ≥ 5(n− 1)/3, then G can be covered by two cycles, or G belongs to an exceptional class.