Abstract: | We present a new condition on the degree sums of a graph that implies the existence of a long cycle. Let c(G) denote the length of a longest cycle in the graph G and let m be any positive integer. Suppose G is a 2-connected graph with vertices x1,…,xn and edge set E that satisfies the property that, for any two integers j and k with j < k, xjxk ? E, d(xi) ? j and d(xk) ? K - 1, we have (1) d(xi) + d(xk ? m if j + k ? n and (2) if j + k < n, either m ? n or d(xj) + d(xk) ? min(K + 1,m). Then c(G) ? min(m, n). This result unifies previous results of J.C. Bermond and M. Las Vergnas, respectively. |