Hunting Snarks

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Hunting snarks Robin Wilson In memory of Martin Gardner Just the place for a Snark! I have said it twice: That alone should encourage the crew. Just the place for a Snark! I have said it thrice: What I tell you three times is true. They sought it with thimbles, they sought it with care; They pursued it with forks and hope; The threatened its life with a railway share; They charmed it with smiles and soap. What is a snark? How many colours are needed to colour the edges of this graph (the ‘Petersen graph’) if edges that meet at a vertex must have different colours? Note: we need at least 3, since each vertex has three edges emerging from it (degree 3) Colouring the Petersen graph We can’t colour it with 3 colours. V. G. Vizing proved that if the largest degree is k, then we always need either k or k + 1 colours. Look for graphs in which every vertex has degree 3, and the edges need 4 colours. Martin Gardner: they’re hard to find, so we’ll call them snarks. The Blanuša snark The Petersen graph with 10 vertices was found in 1898. The Blanusa snark with 18 vertices was found in 1946. (It can be constructed from two Petersen graphs) How long will we have to wait for another one? The Blanuša snark becomes famous Blanche Descartes’s snark (1948) This snark has 210 vertices. The Szekeres snark (1950) This snark with 50 vertices is formed by joining five Petersen graphs. For many years these four snarks were the only ones known. Flower snarks (Grinberg/Isaacs) An infinite family of snarks Goldberg’s snarks By repeating these blocks we get another infinite family of snarks. The double-star snark Small snarks Every snark contains the Petersen graph This was conjectured by Tutte – it’s stronger than the four-colour theorem. A proof was announced by Robertson, Sanders, Seymour and Thomas in 1999 (but remains unpublished). .

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