AbstractWe develop a theory of factorisation of snarks — cubic graphs with edge-chroma-tic number 4 — based on the classical concept of the dot product. Our main concernare irreducible snarks, those where the removal of every nontrivial edge-cut yields a3-edge-colourable graph. We show that if an irreducible snark can be expressed asa dot product of two smaller snarks, then both of them are irreducible. This resultconstitutes the first step towards the proof of the following “unique-factorisation”theorem:Every irreducible snark G can be factorised into a collection {H1, . , Hn} ofcyclically 5-connected irreducible snarks such that G can be reconstructed from themby iterated dot products. Moreover, such a collection is unique up to isomorphismand ordering of the factors regardless of the way in which the decomposition wasperformed.The result is best possible in the sense that it fails for snarks that are closeto being irreducible but themselves are not irreducible. Besides this theor...