6-decomposition of Snarks

A snark is a cubic graph with no proper 3-edge-colouring. In 1996, Nedela and Skoviera 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 ? ( k ) vertices, where the number ? ( k ) only depends on k . The known values of the function ? ( k ) are ? ( 2 ) = 0 , ? ( 3 ) = 1 , ? ( 4 ) = 2 (Goldberg, 1981) 6], and ? ( 5 ) = 5 (Cameron et?al. 1987) 4]. The value ? ( 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 ? ( 6 ) is the last important value to determine. The paper is aimed attacking the problem of determining ? ( 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 ? ( 6 ) ? 20 and there are strong restrictions on the structure of (possibly) missing complements. Part of the proofs are computer assisted.