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 (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 σ k + 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. (1995) [4] proved that if k = 1 and σ 3 ( G ) ≥ n , then G can be covered by two cycles. By these results, we conjecture that if σ 2 k + 1 ( G ) > ( 2 k + 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 a stronger result; if G is 2-connected with σ 5 ( G ) ≥ 5 ( n − 1 ) / 3 , then G can be covered by two cycles, or G belongs to an exceptional class.