A note on the four color theorem

Probably the best strategy to give a computer-free proof of the 4-color theorem is to show that the chromatic polynomial of any planar graph evaluated at 4 is nonzero. After making assumptions like no danglers and adding edges if necessary, a connected graph is a "web" of "interlocking wheels". If C is a web of k interlocking wheels obtained from a planar graph, it is conjectured that its chromatic polynomial P(G, t) = P-1 (t) + ... + P-k (t) with all P-j (4) > 0. In this paper, we prove the conjecture in the base case of the interlocking wheels W-m Lambda(2) W-n for k = 2. As a byproduct, in the last section we apply our result to exponentially simplify the computations to find some interesting facts about a real life graph.