A degree sum condition with connectivity for relative length of longest paths and cycles

For a graph G , p ( G ) and c ( G ) denote the orders of a longest path and a longest cycle of G , respectively. For a graph G , we denote by d G ( x ) and κ ( G ) the degree of a vertex x in G and the connectivity of G , respectively. In this paper, we prove that if G is a 3-connected graph of order n such that ∑ i = 1 4 d G ( x i ) ≥ n + κ ( G ) + 3 for every independent set { x 1 , x 2 , x 3 , x 4 } , then p ( G ) − c ( G ) ≤ 1 . This is a stronger result than the problem of Lu et al. [M. Lu, H. Liu, F. Tian, Two sufficient conditions for dominating cycles, J. Graph Theory 49 (2005) 134–150], and this degree condition is sharp.