Snark (graph theory)

In the mathematical field of graph theory, a snark is a simple, connected, bridgeless cubic graph with chromatic index equal to 4. In other words, it is a graph in which every vertex has three neighbors, and the edges cannot be colored by only three colors without two edges of the same color meeting at a point. (By Vizing's theorem, the chromatic index of a cubic graph is 3 or 4.) In order to avoid trivial cases, snarks are often restricted to have girth at least 5.

Writing in The Electronic Journal of Combinatorics, Miroslav Chladný states that

P. G. Tait initiated the study of snarks in 1880, when he proved that the four color theorem is equivalent to the statement that no snark is planar. The first known snark was the Petersen graph, discovered in 1898. In 1946, Croatian mathematician Danilo Blanuša discovered two more snarks, both on 18 vertices, now named the Blanuša snarks. The fourth known snark was found two years later by Tutte under the pseudonym Blanche Descartes; it has order 210. In 1973, George Szekeres found the fifth known snark — the Szekeres snark. In 1975, Rufus Isaacs generalized Blanuša's method to construct two infinite families of snarks: the flower snark and the BDS or Blanuša–Descartes–Szekeres snark, a family that includes the two Blanuša snarks, the Descartes snark and the Szekeres snark. Isaacs also discovered a 30-vertices snark that does not belong to the BDS family and that is not a flower snark: the double-star snark.

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