My earlier expectation regarding the sufficiency of 4CT to adress this issue, is not correct.
Article An Analytic Proof of Four Color Problem
Article Algebraic Approach to Prove Non-Coplanarity of K9 - publishe...
you don't need to work that hard - see
Computers & Mathematics with Applications
Volume 34, Issue 11, December 1997, Pages 1–8
Biplanar Graphs:: A Survey
L.W. Beineke
the first paragraphs are about k_9
you might also be able to make a counting argument, since k_9 has 36 edges and any decomposition into two parts has at least 18 edges in one or the other part (so you might be able to prove a K_3,3 or K_5 must be in one part or the other)
none of this proves that the 4-color theorem cannot be used, but there are other possibilities
I have already studied, that paper of Beineke. I also gone through the original works of J Battle et. al., and that of Tutte. In this topic, Harary wrote in his book, "No elegant or even reasonable proof is known" in relation to the nonplanarity of K_9.
My latest publication in http://www.ijeijournal.com/pages/v4i6.html may help to give some insight on this issue.
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