6-decomposition of snarks

A snark is a cubic graph with no proper 3-edge-colouring. In 1996, Nedela and Škoviera proved the following theorem: Let G be a snark with an k -edge-cut, k ≥ 2 , whose removal leaves two 3-edge-colourable components M and N . Then both M and N can be completed to two snarks M ̃ and N ̃ of order not exceeding that of G by adding at most κ ( k ) vertices, where the number κ ( k ) only depends on k . The known values of the function κ ( k ) are κ ( 2 ) = 0 , κ ( 3 ) = 1 , κ ( 4 ) = 2 (Goldberg, 1981) [6], and κ ( 5 ) = 5 (Cameron et al. 1987) [4]. The value κ ( 6 ) is not known and is apparently difficult to calculate. In 1979, Jaeger conjectured that there are no 7-cyclically-connected snarks. If this conjecture holds true, then κ ( 6 ) is the last important value to determine. The paper is aimed attacking the problem of determining κ ( 6 ) by investigating the structure and colour properties of potential complements in 6-decompositions of snarks. We find a set of 14 complements that suffice to perform 6 -decompositions of snarks with at most 30 vertices. We show that if this set is not complete to perform 6-decompositions of all snarks, then κ ( 6 ) ≥ 20 and there are strong restrictions on the structure of (possibly) missing complements. Part of the proofs are computer assisted.