Three-coloring triangle-free graphs on surfaces IV. Bounding face sizes   of 4-critical graphs
  Let G be a 4-critical graph with t triangles, embedded in a surface of genus g. Let c be the number of 4-cycles in G that do not bound a 2-cell face. We prove that the sum of lengths of (>=5)-faces of G is at most linear in g+t+c-1. 
Zdenek Dvorak and Daniel Kral and Robin Thomas
math.CO