On Vizing's problem for triangle-free graphs

Abstract

We prove that $\ddot{I}‡(G)\le \lceil (\hat{I}”+1)/2\rceil +1$ for any triangle-free graph $G$ of maximum degree $\hat{I}”$ provided $\hat{I}”\ge 524$. 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.

11 pages; we learned from Z Dvorak and R Wdowinski that a conjecture we posed was known, so Sec 3 in v3 is adapted accordingly, and the suggestions of referees are also incorporated