Degree sums and dominating cycles

Abstract A cycle C of a graph G is dominating if any vertex of V ( G ) ∖ V ( C ) has at least one neighbor on C and V ( G ) ∖ V ( C ) is an independent set. Let G be a k -connected graph of order n ≥ 3 with k ≥ 2 . In this paper, we show that every longest cycle of G is dominating if the degree sums is more than ( k + 1 ) ( n + 1 ) ∕ 3 for any k + 1 pairwise nonadjacent vertices, and the lower bound is sharp, which generalizes the results due to Bondy (1980) for k = 2 and Lu et al. (2005) for k = 3 .