Relative length of longest paths and cycles in 3-connected graphs

For a graph G, let p(G) denote the order of a longest path in G and c(G) th e order of a longest cycle in G, respectively. We show that if G is a 3-con nected graph of order n such that Sigma (4)(i=1) deg(G) x(i) greater than o r equal to 3/2 n+1 for every independent set {x(1), x(2), x(3), x(4)}, then G satisfies c(G) greater than or equal to p(G) - 1. Using this result, we give several lower bounds to the circumference of a 3-connected graph. (C) 2001 John Wiley & Sons, Inc.