Three-coloring triangle-free graphs on surfaces III. Graphs of girth   five
  We show that the size of a 4-critical graph of girth at least five is bounded by a linear function of its genus. This strengthens the previous bound on the size of such graphs given by Thomassen. It also serves as the basic case for the description of the structure of 4-critical triangle-free graphs embedded in a fixed surface, presented in a future paper of this series. 
Zden\v{e}k Dvo\v{r}\'ak and Daniel Kr\'a\v{l} and Robin Thomas
math.CO