A Degree Sum Condition Concerning the Connectivity and the Independence Number of a Graph

Let G be a graph and S. V(G). We denote by a(S) the maximum number of pairwise nonadjacent vertices in S. For x, y epsilon V(G), the local connectivity kappa(x, y) is defined to be the maximum number of internally-disjoint paths connecting x and y in G. We define kappa(S) = min{kappa(x, y) : x, y epsilon S, x not equal y}. In this paper, we show that if kappa(S) >= 3 and Sigma(4)(i=1) dG(x(i)) >= | V(G)|+ kappa(S)+ alpha(S) - 1 for every independent set {x(1), x(2), x(3), x(4)} subset of S, then G contains a cycle passing through S. This degree condition is sharp and this gives a new degree sum condition for a 3-connected graph to be hamiltonian.