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A note on the four color theorem

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Abstract

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 G is a web of k interlocking wheels obtained from a planar graph, it is conjectured that its chromatic polynomial \(P(G,t) = P_1(t)+ \cdots + P_k(t)\) with all \(P_j(4) >0\). In this paper, we prove the conjecture in the base case of the interlocking wheels \(W_m\wedge _2W_n\) 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.

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Acknowledgements

The authors would like to thank the College of Mathematical and Physical Sciences of BYU for financially supporting this research.

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Authors and Affiliations

  1. Department of Mathematics Education, BYU, Provo, UT, 84602, USA

    Jennifer Canizales

  2. Department of Mathematics, BYU, Provo, UT, 84602, USA

    Jasbir S. Chahal

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  1. Jennifer Canizales
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  2. Jasbir S. Chahal
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Correspondence to Jasbir S. Chahal.

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Canizales, J., Chahal, J.S. A note on the four color theorem. Aequat. Math. 97, 1–11 (2023). https://doi.org/10.1007/s00010-022-00929-8

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  • DOI: https://doi.org/10.1007/s00010-022-00929-8

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