Triangle-free planar graphs with the smallest independence number
  Steinberg and Tovey proved that every n-vertex planar triangle-free graph has an independent set of size at least (n+1)/3, and described an infinite class of tight examples. We show that all n-vertex planar triangle-free graphs except for this one infinite class have independent sets of size at least (n+2)/3. 
Zden\v{e}k Dvo\v{r}\'ak and Tom\'a\v{s} Masa\v{r}\'ik and Jan   Mus\'ilek and Ond\v{r}ej Pangr\'ac
math.CO cs.DM