Wheels in planar graphs and Hajós graphs

It was conjectured by Hajós that graphs containing no K 5 -subdivision are 4-colorable. Previous results show any possible minimum counterexample to Hajós' conjecture, called graph, is 4-connected but not 5-connected. In this paper, we if a graph admits 4-cut or 5-cut with planar side then the must be small contains special wheel. This step in our effort reduce conjecture Four Color Theorem.