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Tutte
In 1884 P.G. Tait conjectured that every polyhedral graph has a Hamilton cycle [24], and thus began one of the most celebrated stories in the mathematical theory of graphs. If true, Tait’s conjecture would imply the famous Four Colour Theorem; a series of attempts to prove Tait’s conjecture consumed Graph Theorists for the next 60 years. The conjecture was finally disproved by W. T. Tutte, using the graph in these pictures. The pictures clearly show an automorphism group of the graph as rotational symmetries. The nodes are clustered in the picture by orbits (i.e., equivalence classes under automorphism), and each orbit is shown by the background dot colours (in a weighted Voronoi tessellation). In between the outer face and the central node are three mutually isomorphic subgraphs, called “Tutte fragments”. A simple mathematical argument, based on the orbits and the “Tutte fragments”, shows that the graph has no Hamilton cycle (see Tutte’s 1946 paper “On Hamiltonian circuits”, in the Journal of the London Mathematical Society,1946).