Résumé
We prove that chi(G) <= inverted right perpendicular(triangle+1)/2inverted left perpendicular +1 for any triangle-free graph G of maximum degree triangle provided triangle >= 524. This gives tangible progress towards an old problem of Vizing, in a form cast by Reed. We use a method of Hurley and Pirot, which in turn relies on a new counting argument of the second author.