Color fixation and color identity in 4-chromatic graphs
  I argue that there is no 4-chromatic planar graph with a joinable pair of color identical vertices, i.e., given a 4-chromatic planar graph G and a pair of vertices {u, v} in G, if the color of u equals the color of v in every 4-coloring of G, then there is no planar supergraph of G where u and v are adjacent. This is equivalent to the Four Color Theorem. (My argument is a variation of my argument in arXiv:1402.7368) 
Asbj{\o}rn Br{\ae}ndeland
math.GM