On Vizing's problem for triangle-free graphs

We prove that $\chi (G)\le \lceil (\mathrm{\Delta }+1)\mathrm{/}2\rceil +1$ for any triangle-free graph $G$ of maximum degree $\mathrm{\Delta }$ provided $\mathrm{\Delta }\ge 524$. 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.