A SHARP LOWER-BOUND FOR THE CIRCUMFERENCE OF 1-TOUGH GRAPHS WITH LARGE DEGREE SUMS

We show that every 1-tough graph G on n greater than or equal to 3 vertices with sigma(3) greater than or equal to n has a cycle of length at least min{n, n + (sigma(3)/3) - alpha + 1}, where sigma(3) denotes the minimum value of the degree sum of any 3 pairwise nonadjacent vertices and alpha the cardinality of a maximum independent set of vertices in G. Our inequality is sharp and implies some sufficient conditions of hamiltonian cycles. (C) 1995 John Wiley & Sons, Inc.