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Irreducible snarks of given order and cyclic connectivity. (English) Zbl 1092.05026
Summary: 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 $$\geqslant 210$$. (Cyclically 7-connected snarks are believed not to exist.)

##### MSC:
 05C15 Coloring of graphs and hypergraphs
edge-colouring
Full Text:
##### References:
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