The problem of establishing the number of perfect matchings necessary to cover the edge-set of a cubic bridgeless graph is strictly related to a famous conjecture of Berge and Fulkerson. In this paper we prove that deciding whether this number is at most 4 for a given cubic bridgeless graph is NP-complete. We also construct an infinite family F of snarks (cyclically 4-edge-connected cubic graph...

Subversion zero knowledge for non-interactive proof systems demands that zero knowledge (ZK) be maintained even when the common reference string (CRS) is chosen maliciously. SNARKs are proof systems with succinct proofs, which are at the core of the cryptocurrency Zcash, whose anonymity relies on ZK-SNARKs; they are also used for ZK contingent payments in Bitcoin. We show that under a plausible...

We consider a logicist approach to natural language understanding based on the translation of a quasi-logical form into a temporal logic, explicitly constructed for the representation of action and change, and the subsequent reasoning about this semantic structure in the context of a background knowledge theory using automated theorem proving techniques. The approach is substantiated through a ...

We present a proof system that allows efficient verification of NP statements, given proofs produced by an untrusted yet computationally-bounded prover. Our system is publicly verifiable: after a trusted third-party has generated a proving key and a verification key, anyone can use the proving key to generate non-interactive proofs for adaptively-chosen NP statements, and the proofs can be veri...

Non-interactive zero-knowledge proofs, and more specifically succinct non-interactive arguments (zk-SNARKs), have been proven to be the “Swiss army knife” of blockchain distributed ledger space, with a variety applications in privacy, interoperability scalability. Many commonly used SNARK systems rely on structured reference string, secure generation which turns out their Achilles heel: If rand...

For many of the unsolved problems concerning cycles and matchings in graphs it is known that it is su cient to prove them for snarks, the class of nontrivial 3-regular graphs which cannot be 3-edge coloured. In the rst part of this paper we present a new algorithm for generating all non-isomorphic snarks of a given order. Our implementation of the new algorithm is 14 times faster than previous ...

In 1969, Grünbaum conjectured that snarks do not have polyhedral embeddings into orientable surfaces. To describe the deviation from polyhedrality, we define the defect of a graph and use it to study embeddings of superpositions of cubic graphs into orientable surfaces. Superposition was introduced in [4] to construct snarks with arbitrary large girth. It is shown that snarks constructed in [4]...

The well-known 5-flow Conjecture of Tutte, stated originally for integer flows, claims that every bridgeless graph has circular flow number at most 5. It is a classical result that the study of the 5-flow Conjecture can be reduced to cubic graphs, in particular to snarks. However, very few procedures to construct snarks with circular flow number 5 are known. In the first part of this paper, we ...