Cycles and paths in triangle-free graphs

Abstract

Let G be a triangle-free graph of order n and minimum degree δ > n ∕ 3. We will determine all lengths of cycles occurring in G. In particular, the length of a longest cycle or path in G is exactly the value admitted by the independence number of G. This value can be computed in time O(n 2.5) using the matching algorithm of Micali and Vazirani. An easy consequence is the observation that triangle-free non-bipartite graphs with δ≥ n are hamiltonian.