Construction of permutation snarks

Permutation snarks

A permutation snark is a cubic graph with no 3-edge-colouring that contains a 2-factor consisting of two induced circuits. In the talk we analyse the basic properties of permutation snarks, focusing on the structure of edge-cuts of size 4 and 5. As an application of our knowledge we provide rich families of cyclically 4edge-connected and 5-edge-connected permutation snarks of order 8n+2 for eac...

A New Construction of Permutation Arrays

and a positive integer m. The resulted array has ( |PA(p, k)| · p(m−1)(p−k) )m rows. Compared to the other constructions, the new construction gives a permutation array of far bigger size with a large minimum distance, for example, when k ≥ 2p/3. Moreover the proposed construction provides an algorithm to find the i-th row of PA ( pm, pm−1k ) for a given index i very simply. key words: permutat...

A Construction of Endo-permutation Modules

Treelike Snarks

In this talk we present snarks whose edges cannot be covered by fewer than five perfect matchings. Esperet and Mazzuoccolo found an infinite family of such snarks, generalising an example provided by Hägglund. We construct another infinite family, arising from a generalisation in a different direction. The proof that this family has the requested property is computer-assisted. In addition, we p...

Factorisation of Snarks

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 irreducib...