Irreducible snarks of given order and cyclic connectivity

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 210. (Cyclically 7-connected snarks are believed not to exist.)

On hypohamiltonian snarks and a theorem of Fiorini

“…We then prove a strengthening of Steffen's theorem, which is best possible, as all orders for which hypohamiltonian snarks exist are determined. Our result is stronger than a theorem of Máčajová and Škoviera [15] in the sense that our result implies theirs, while the inverse implication does not hold. Finally, in Section 4 we comment upon and verify a conjecture of Steffen on hypohamiltonian snarks [22] for small hypohamiltonian snarks.…”

“…We shall now prove a strengthening of Steffen's theorem, which in a sense is strongest possible since we will determine all orders for which hypohamiltonian snarks exist. We emphasise that our proof's mechanism contains significantly fewer "moving parts" than Máčajová and Škoviera's [15], and, as mentioned in the introduction, our theorem also strengthens their result. We do need the following two easily verifiable lemmas.…”

“…The hamiltonian paths for the other values of v can be found in the Appendix. v = 0: 12,13,14,15,32,31,30,29,6,5,10,9,8,7,2,1,23,24,25,26,4,3,28,27,35,34,33,18,19,20,21,22,17,16,11 Since Steffen's statement of Lemma 3.3 remains intact, the proof and statement of his main result, reproduced above as Theorem 3.2, are correct as given in [21]. Even though we prove a stronger version of Steffen's theorem in the next section, we think it is important to fix the proof of Lemma 3.3 as there may be others who rely on this lemma, or might want to rely on it in the future.…”

“…Figure 4 shows the second Blanuša snark B 2 . By computer we determined that B 2 has exactly three pairs of suitable edges: ((6, 8), (10,16)), ((3, 9), (12,17)) and ((4, 7), (13,15)). We will now prove by hand that ((6, 8), (10,16)) is a suitable edge pair.…”

Coverings of cubic graphs and 3-edge colorability

“…It follows from the definition that no loop cannot serve as a semi-edge of M . Semi-edges are usually grouped into pairwise disjoint connectors [7,9]. A multipole with k semi-edges is called k-pole.…”