Bipartizing fullerenes
  A fullerene graph is a cubic bridgeless planar graph with twelve 5-faces such that all other faces are 6-faces. We show that any fullerene graph on n vertices can be bipartized by removing O(sqrt{n}) edges. This bound is asymptotically optimal. 
Zdenek Dvorak and Bernard Lidicky and Riste Skrekovski
math.CO