On triangle-free graphs that do not contain a subdivision of the   complete graph on four vertices as an induced subgraph
  We prove a decomposition theorem for the class of triangle-free graphs that do not contain a subdivision of the complete graph on four vertices as an induced subgraph. We prove that every graph of girth at least~5 in this class is 3-colorable. 
Nicolas Trotignon and Kristina Vu\v{s}kovi\'c
math.CO