Irreducible snarks of given order and cyclic connectivity

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Abstract

A snark is a "nontrivial" cubic graph whose edges cannot be properly coloured by three colours; it is irreducible if each nontrivial edge-cut divides the snark into colourable components. Irreducible snarks can be viewed as simplest uncolourable structures. In fact, all snarks can be composed from irreducible snarks in a suitable way. In this paper we deal with the problem of the existence of irreducible snarks of given order and cyclic connectivity. We determine all integers n for which there exists an irreducible snark of order n, and construct irreducible snarks with cyclic connectivity 4 and 5 of all possible orders. Moreover, we construct cyclically 6-connected irreducible snarks of each even order {greater than or slanted equal to} 210. (Cyclically 7-connected snarks are believed not to exist.). © 2006 Elsevier B.V. All rights reserved.

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Máčajová, E., & Škoviera, M. (2006). Irreducible snarks of given order and cyclic connectivity. Discrete Mathematics, 306(8–9), 779–791. https://doi.org/10.1016/j.disc.2006.02.003

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