A note on generalized pentagons

Thomassen introduced the notion of a generalized pentagon and proved that the chromatic number of a triangle-free graph with n vertices and minimum degree at least c n , c > 1 3 , is at most 2 ( 3 c − 1 ) − ( 4 c − 1 ) / ( 3 c − 1 ) , the first bound independent of the order n . We present a short proof of the stronger upper bound ( 3 c − 1 ) − 1 , again based on generalized pentagons.