We discuss the construction of snarks (that is, cyclically 4-edge connected cubic graphs of girth at least five which are not 3-edge colourable) by using what we call colourable snark units and a welding process.

In 2003, Cavicchioli et al. corrected an omission in the statement and proof of Fiorini’s theorem from 1983 on hypohamiltonian snarks. However, their version of this theorem contains an unattainable condition for certain cases. We discuss and extend the results of Fiorini and Cavicchioli et al. and present a version of this theorem which is more general in several ways. Using Fiorini’s erroneou...

Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Conjecture. In 2003, Cavicchioli et al. proved that for snarks with less than 30 vertices, the total chromatic number is 4, and proposed the problem of finding (if any) the smallest snark which is not 4-total colorable. Several families of snarks have had their total...

Since the 19th century, much academic effort has been expended researching leadership. Bodies of theory have risen to dominance, proved unsatisfactory and replaced by another generation ultimately disappointing leadership thought. This repetitive pattern continues, so we ask what motivates this continuing, seemingly fruitless search? Focusing on researchers not per se, our analysis is inspired ...

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...

The paper discusses the basic principles of construction and main types zeroknowledge succinct non-interactive argument knowledge (zk-SNARK) which is used in model a three-way insecure computing environment based on sets polynomials. A number zk-SNARK cryptographic protocols with different algorithms for generating public parameters (Trusted Setup) are given, constructing proofs reliability cal...

We determine the circular chromatic index of flower snarks, by showing that χc(F3) = 7/2, χ ′ c(F5) = 17/5 and χ ′ c(Fk) = 10/3 for every odd integer k ≥ 7, where Fk denotes the flower snark on 4k vertices.