Relative length of longest paths and longest cycles in triangle-free graphs

In this paper, we study triangle-free graphs. Let G = (V, E) be an arbitrary triangle-free graph with minimum degree at least two and sigma(4)(G) >= vertical bar V (G)vertical bar + 2. We first show that either for any path P in G there exists a cycle C such that vertical bar V-P \ V-C vertical bar <= 1, or G is isomorphic to exactly one exception. Using this result, we show that for any set S of at most 6 vertices in G there is a cycle C such that S subset of V-C. (C) 2007 Elsevier B.V. All rights reserved.