# Factorisation of Snarks

### Abstract

We develop a theory of factorisation of snarks — cubic graphs with edge-chromatic number $4$ — based on the classical concept of the dot product. Our main concern are *irreducible* snarks, those where the removal of every nontrivial edge-cut yields a $3$-edge-colourable graph. We show that if an irreducible snark can be expressed as a dot product of two smaller snarks, then both of them are irreducible. This result constitutes the first step towards the proof of the following "unique-factorisation" theorem:

*Every irreducible snark $G$ can be factorised into a collection $\{H_1,\dots,H_n\}$ of cyclically $5$-connected irreducible snarks such that $G$ can be reconstructed from them by iterated dot products. Moreover, such a collection is unique up to isomorphism and ordering of the factors regardless of the way in which the decomposition was performed.*

The result is best possible in the sense that it fails for snarks that are close to being irreducible but themselves are not irreducible. Besides this theorem, a number of other results are proved. For example, the unique-factorisation theorem is extended to the case of factorisation with respect to a preassigned subgraph $K$ which is required to stay intact during the whole factorisation process. We show that if $K$ has order at least $3$, then the theorem holds, but is false when $K$ has order $2$.