Almost all graphs with high girth and suitable density have high chromatic number

Erdös proved that there exist graphs of arbitrarily high girth and arbitrarily high chromatic number. We give a different proof (but also using the probabilistic method) that also yields the following result on the typical asymptotic structure of graphs of high girth: for all ℓ ≥ 3 and k ∈ ℕ there exist constants C 1 and C 2 so that almost all graphs on n vertices and m edges whose girth is greater than ℓ have chromatic number at least k , provided that $C_{1}n\,\leq\, m\,\leq\, C_{2}n^{\ell /(\ell -1)}$ . © 2001 John Wiley & Sons, Inc. J Graph Theory 37: 220–226, 2001