These might make sense to you if you're familiar with four-color-theorem proofs; I don't lay out the arguments here. These pages each examine whether a certain subgraph of a minimal counterexample is reducible.
(Wilson
is Wilson, Robert A., Graphs, Colourings and the Four-colour Theorem, 2002)
Sincerely, Izzy Dupree, Elias Zeidan and Sam Auciello.