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The well-known shortest cycle cover conjecture suggests that every bridgeless graph $G$ can have its edges covered with a collection of cycles of total length not exceeding $\frac75\cdot|E(G)|$. Th...

The perfect matching index of a cubic graph $G$, denoted by $\pi(G)$, is the smallest number of perfect matchings that cover all the edges of $G$. According to the Berge--Fulkerson conjecture, $\pi...

We consider the weakly convex and convex domination numbers for two classes of graphs: generalized Petersen graphs and flower snark graphs. For a given generalized Petersen graph GP(n, k), we prove that if k = 1 and n ≥ 4 then both the weakly convex domination number γwcon(GP(n, k)) and the convex domination number γcon(GP(n, k)) are equal to n. For k ≥ 2 and n ≥ 13, γwcon(GP(n, k)) = γcon(GP(n, k)) = 2n, which is the order of GP(n, k). Special cases for smaller graphs are solved by the exact method. For a flower snark graph Jn, where n is odd and n ≥ 5, we prove that γwcon(Jn) =2n and γcon(Jn) = 4n.

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 summarise some of these methods and we propose new ones based on variations of the known constructions. Afterwards, we prove that all such methods are nothing but particular instances of a more general construction that we introduce into detail. In the second part, we consider many instances of this general method and we determine when our method permits to obtain a snark with circular flow number 5. Finally, by a computer search, we determine all snarks having circular flow number 5 up to 36 vertices. It turns out that all such snarks of order at most 34 can be obtained by using our method, and that the same holds for 96 of the 98 snarks of order 36 with circular flow number 5., 27 pages; submitted for publication

The Berge-Fulkerson conjecture, originally formulated in the language of mathematical programming, asserts that the edges of every bridgeless cubic ( 3 3 -valent) graph can be covered with six perfect matchings in such a way that every edge is in exactly two of them. As with several other classical conjectures in graph theory, every counterexample to the Berge-Fulkerson conjecture must be a non- 3 3 -edge-colorable cubic graph. In contrast to Tutte’s 5-flow conjecture and the cycle double conjecture, no nontrivial reduction is known for the Berge-Fulkerson conjecture. In the present paper, we prove that a possible minimum counterexample to the conjecture must be cyclically 5 5 -edge-connected.

Brands have long been advised to not post or encourage negative messages about competing brands; but individual users independently posting “snark” or celebrating the travails of competing brands v...

Professor John Agnew makes a convincing case for the meaninglessness of Brexit, a vain attempt to recreate a kind of nation-state that never existed. It is perhaps ironic that this project should h...

Motivated by a conjecture of Grunbaum and a problem of Katona, Kostochka, Pach, and Stechkin, both dealing with non-Hamiltonian n-vertex graphs and their (n - 2)-cycles, we investigate K_2-Hamiltonian graphs, i.e., graphs in which the removal of any pair of adjacent vertices yields a Hamiltonian graph. In this first part, we prove structural properties and show that there exist infinitely many cubic non-Hamiltonian K_2-Hamiltonian graphs, both of the 3-edge-colorable and the non-3-edge-colorable variety. In fact, cubic K_2-Hamiltonian graphs with chromatic index 4 (such as Petersen's graph) are a subset of the critical snarks. On the other hand, it is proven that non-Hamiltonian K_2-Hamiltonian graphs of any maximum degree exist. Several operations conserving K_2-Hamiltonicity are described, one of which leads to the result that there exists an infinite family of non-Hamiltonian K_2-Hamiltonian graphs in which, asymptotically, a quarter of vertices has the property that removing such a vertex yields a non-Hamiltonian graph. We extend a celebrated result of Tutte by showing that every planar K_2-Hamiltonian graph with minimum degree at least 4 is Hamiltonian. Finally, we investigate K_2-traceable graphs and discuss open

A well known conjecture of Alon and Tarsi (1985) states that every bridgeless graph admits a cycle cover of length not exceeding \(\frac{7}{5}\cdot m\), where m is the number of edges. Although there exist infinitely many cubic graphs with covering ratio 7/5, there is an extensive evidence that most cyclically 4-edge-connected cubic graphs have covering ratio close to the natural lower bound of 4/3. In line with this observation, Brinkmann et al. (2013) proposed a conjecture that every cyclically 4-edge-connected cubic graph has a cycle cover of length at most \(\frac{4}{3} m+o(m)\). In this paper we disprove the conjecture.

Polynomial commitment schemes (PCS) have recently been in the spotlight for their key role in building SNARKs. A PCS provides the ability to commit to a polynomial over a finite field and prove its evaluation at points. A succinct PCS has commitment and evaluation proof size sublinear in the degree of the polynomial. An efficient PCS has sublinear proof verification. Any efficient and succinct PCS can be used to construct a SNARK with similar security and efficiency characteristics (in the random oracle model).

International audience; A zero-knowledge proof is a method by which one can prove knowledge of general non-deterministic polynomial (NP) statements. SNARKs are in addition non-interactive, short and cheap to verify. This property makes them suitable for recursive proof composition, that is proofs attesting to the validity of other proofs. To achieve this, one moves the arithmetic operations to the exponents. Recursive proof composition has been empirically demonstrated for pairing-based SNARKs via tailored constructions of expensive pairing-friendly elliptic curves namely a pair of 753-bit MNT curves, so that one curve's order is the other curve's base field order and vice-versa. The ZEXE construction restricts to one layer proof composition and uses a pair of curves, BLS12-377 and CP6-782, which improve significantly the arithmetic on the first curve. In this work we construct a new pairing-friendly elliptic curve to be used with BLS12-377, which is STNFS-secure and fully optimized for one layer composition. We propose to name the new curve BW6-761. This work shows that it is at least five times faster to verify a composed SNARK proof on this curve compared to the previous state-of-the-art, and proposes an optimized Rust implementation that is almost thirty times faster than the one available in ZEXE library.

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. An infinite family R , of cyclically 5-edge-connected rotation snarks, was discovered in [European J. Combin. 2021] by Macajova and Skoviera. In this paper, the Berge-Fulkerson conjecture is verified for the family R , and furthermore, a sup-family of R . Catlin’s contractible configuration and Tutte’s integer flow are applied here as the key methods.

In the late nineteenth century, the British writer Lewis Carroll published a nonsensical poem calledThe Hunting of the Snarkin which an unlikely alliance hunts a fictional animal, which Carroll named the “snark.” Despite the alliance's intense search for the snark and their questions about how to describe and classify it (apparently, “a Boojum”), they do not find it. I want to suggest that any effort to locate “emotions” in the Hebrew Bible or the ancient Near East is comparable to hunting the snark. If we want our hunt to be successful, we will turn away from “the emotions” and toward something more like the psychological taxonomy that the emotions displaced in the late-modern period: namely, the taxonomy of “passions and affections.” “The emotions” are simply not to be found in the Hebrew Bible or in the historical contexts behind its emergence.

We discuss an omission in the statement and proof of Fiorini's 1983 theorem on hypohamiltonian snarks and present a version of this theorem which is more general in several ways. Using Fiorini's erroneous result, Steffen showed that hypohamiltonian snarks exist for some $n \ge 10$ and each even $n \ge 92$. We rectify Steffen's proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffen's theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exists. This also strengthens a result of M\'{a}\v{c}ajov\'{a} and \v{S}koviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices., Comment: 21 pages; submitted for publication

Objective. The aim of the present study was to examine the effect of chromium disilicide and titanium nitride nanoparticles on the expression level of genes encoding important regulatory factors (IGFBP1, IGFBP2, IGFBP3, IGFBP4, IGFBP5, SNARK/NUAK2, CD36, and PECAM1/CD31) in mouse liver for evaluation of possible toxic effects of these nanoparticles. Methods. Male mice received 20 mg chromium disilicide nanoparticles (45 nm) and titanium nitride nanoparticles (20 nm) with food every working day for 2 months. The expression of IGFBP1, IGFBP2, IGFBP3, IGFBP4, IGFBP5, SNARK, CD36, and PECAM1 genes in mouse liver was studied by quantitative polymerase chain reaction. Results. Treatment of mice with chromium disilicide nanoparticles led to down-regulation of the expression of IGFBP2, IGFBP5, PECAM1, and SNARK genes in the liver in comparison with control mice, with more prominent changes for SNARK gene. At the same time, the expression of IGFBP3 and CD36 genes was increased in mouse liver upon treatment with chromium disilicide nanoparticles. We have also shown that treatment with titanium nitride nanoparticles resulted in down-regulation of the expression of IGFBP2 and SNARK genes in the liver with more prominent changes for SNARK gene. At the same time, the expression of IGFBP3, IGFBP4, and CD36 genes was increased in the liver of mice treated with titanium nitride nanoparticles. Furthermore, the effect of chromium disilicide nanoparticles on IGFBP2 and CD36 genes expression was significantly stronger as compared to titanium nitride nanoparticles. Conclusions. The results of this study demonstrate that chromium disilicide and titanium nitride nanoparticles have variable effects on the expression of IGFBP2, IGFBP3, IGFBP4, IGFBP5, SNARK, CD36, and PECAM1 genes in mouse liver, which may reflect the genotoxic activities of the studied nanoparticles.

In April of 1907, after more than six months of setbacks and delays while their boat was under construction, Charmian was relieved to get Jack onto their new ketch, the Snark, away from hard-drinki...

Among various NIZK arguments, zk-SNARKs are the most efficient constructions in terms of proof size and verification which are two critical criteria for large scale applications. Currently, Groth’s construction, \(\textsf {Groth16}\), from Eurocrypt’16 is the most efficient and widely deployed one. However, it is proven to achieve only knowledge soundness, which does not prevent attacks from the adversaries who have seen simulated proofs. There has been considerable progress in modifying \(\textsf {Groth16}\) to achieve simulation extractability to guarantee the non-malleability of proofs. We revise the Simulation Extractable (SE) version of \(\textsf {Groth16}\) proposed by Bowe and Gabizon that has the most efficient prover and \(\mathsf {crs}\) size among the candidates, although it adds Random Oracle (RO) to the original construction. We present a new version which requires 4 parings in the verification, instead of 5. We also get rid of the RO at the cost of a collision resistant hash function and a single new element in the \(\mathsf {crs}\). Our construction is proven in the generic group model and seems to result in the most efficient SE variant of \(\textsf {Groth16}\) in most dimensions.

In this essay, the author notes that for the past half-century, psychologists have examined how humans make use of spatial representations when making judgments about numerical properties of sets of items. This line of work was initiated by Frank Restle (1970), who asked college students at Indiana University to choose the larger number, either the sum of A + B or C, as rapidly as possible. Restle found that the timing of people's choices fit an analog model of numerical judgment that had been proposed a few years earlier (Moyer & Landauer, 1967) and that people seemed to judge the magnitude of numbers by their position on a mental number line. The hypothetical number line is now called the "mental number line," and the effect on latency to respond to questions about numbers that results from use of the mental number line has been labeled the "Spatial Numerical Association of Response Codes" or SNARC for short. The author notes that she has parodied Frank Beach's (1950) title for his classic essay about the state of comparative psychology at the mid-20th century for the title of this piece. Beach, in turn, was inspired by Lewis Carroll's nonsense poem, " The Hunting of the Snark," published in 1876. The poem chronicles "the impossible voyage of an improbable crew to find an inconceivable creature," the snark. According to the poem, most species of snarks are relatively harmless. The boojum, however, is a dangerous snark, because those who catch sight of a boojum "suddenly vanish away." Here, the author wants to turn the meaning of the homonym SNARC a bit to suggest that the SNARC might itself vanish, perhaps to metamorphose into a more complex entity. (PsycINFO Database Record (c) 2019 APA, all rights reserved).

A graph is SSP (Super Strongly Perfect) if all of its (induced) subgraph H in G obsesses a (minimal dominating set) MDS that link up all of its cliques (maximal) in H. In this paper, SSP Structure along with its parameters (counting of maximal cliques, cardinality of dominating set (Minimal) and colourability of closed Helm graph is analysed and also Flower snark graph families are investigated.

We describe two new algorithms for the generation of all non-isomorphic cubic graphs with girth at least k≥5 that are very efficient for 5≤k≤7 and show how these algorithms can be restricted to generate snarks with girth at least k. Our implementation of these algorithms is more than 30, respectively 40 times faster than the previously fastest generator for cubic graphs with girth at least six and seven, respectively. Using these generators we have also generated all nonisomorphic snarks with girth at least six up to 38 vertices and show that there are no snarks with girth at least seven up to 42 vertices. We present and analyze the new list of snarks with girth 6.

An r-dynamic coloring of a graph is a proper minimum coloring of the vertices such that | c ( N ( υ ) ) | ≥ min { r , d e g G ( υ ) } , for each υ ∈ V ( G ) and it is denoted by χ r ( G ) . Snark are bridgeless cubic connected graph in which every vertex has three neighbors. In this paper, we shown the r-dynamic coloring for Celmins-swart snark, Double star snark, Loupekine snark, Szekeres snark, Watkins snark and infinite family of flower snark with its special case as Tietze’s graph. This result reinforce that most of the snark are 6-colorable for its maximum degree and also we give the procedures to construct a r-dynamic coloring of each snark.

The family of snarks -- connected bridgeless cubic graphs that cannot be 3-edge-coloured -- is well-known as a potential source of counterexamples to several important and long-standing conjectures in graph theory. These include the cycle double cover conjecture, Tutte's 5-flow conjecture, Fulkerson's conjecture, and several others. One way of approaching these conjectures is through the study of structural properties of snarks and construction of small examples with given properties. In this paper we deal with the problem of determining the smallest order of a nontrivial snark (that is, one which is cyclically 4-edge-connected and has girth at least 5) of oddness at least 4. Using a combination of structural analysis with extensive computations we prove that the smallest order of a snark with oddness at least 4 and cyclic connectivity 4 is 44. Formerly it was known that such a snark must have at least 38 vertices [J. Combin. Theory Ser. B 103 (2013), 468--488] and one such snark on 44 vertices was constructed by Lukot'ka et al. [Electron. J. Combin. 22 (2015), #P1.51]. The proof requires determining all cyclically 4-edge-connected snarks on 36 vertices, which extends the previously compiled list of all such snarks up to 34 vertices [J. Combin. Theory Ser. B, loc. cit.]. As a by-product, we use this new list to test the validity of several conjectures where snarks can be smallest counterexamples., 21 pages

Jack London musste in seinem abenteuerlichen Leben manche Krankheit uberstehen. Bei seiner Reise im Sudpazifik bekamen er und die anderen Crew-Mitglieder, neben anderem, die Frambosie, auch „Himbeerpocken“ genannt.

The origin of life and the existence of extraterrestrial life are among the most important scientific puzzles of our time. Humans have embarked on a great quest, the discovery of extraterrestrial life: We are seeking the Snark. The work of characterizing celestial bodies within our solar system and beyond contributes toward solving the puzzle, and future astronomical observations of exoplanets are also essential. However, in promoting those efforts or the choice of targets, excessive emphasis on habitability is bad science policy: It presumes a greater understanding of the concept than we currently possess, and it may fail to recognize the extraordinary diversity of planetary environments.

The signaling mechanisms mediating myocardial glucose transport are not fully understood. Sucrose nonfermenting AMP-activated protein kinase (AMPK)-related kinase (SNARK) is an AMPK-related protein kinase that is expressed in the heart and has been implicated in contraction-stimulated glucose transport in mouse skeletal muscle. We first determined if SNARK is phosphorylated on Thr208 , a site critical for SNARK activity. Mice were treated with exercise, ischemia, submaximal insulin, or maximal insulin. Treadmill exercise slightly, but significantly increased SNARK Thr208 phosphorylation. Ischemia also increased SNARK Thr208 phosphorylation, but there was no effect of submaximal or maximal insulin. HL1 cardiomyocytes were used to overexpress wild-type (WT) SNARK and to knockdown endogenous SNARK. Overexpression of WT SNARK had no effect on ischemia-stimulated glucose transport; however, SNARK knockdown significantly decreased ischemia-stimulated glucose transport. SNARK overexpression or knockdown did not alter insulin-stimulated glucose transport or glycogen concentrations. To study SNARK function in vivo, SNARK heterozygous knockout mice (SNARK+/- ) and WT littermates performed treadmill exercise. Exercise-stimulated glucose transport was decreased by ~50% in hearts from SNARK+/- mice. In summary, exercise and ischemia increase SNARK Thr208 phosphorylation in the heart and SNARK regulates exercise-stimulated and ischemia-stimulated glucose transport. SNARK is a novel mediator of insulin-independent glucose transport in the heart.

A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. The Berge–Fulkerson conjecture proposed in 1971 states that every bridgeless cubic graph contains a family of six perfect matchings such that each edge is contained in exactly two of them. This conjecture holds trivialy for 3-edge colorable graphs. Thus a possible minimum counterexample for the conjecture is a snark. In this paper we shown that the conjecture holds for families of snarks such as Loupekhine snarks of first and second kind and the Watkins snark. We also determine the excessive index for Loupekhine snarks of first and second kind.

Neste trabalho, provamos que todos os membros de uma familia infinita de snarks obtida pela superposicao, definida por Kochol em 1996, sao Tipo 1. Este resultado contribui com as evidencias de que nao existe snark Tipo 2 com cintura pelo menos 5.

It is well-known that a 2-edge-connected cubic graph has a 3-edge-colouring if and only if it has a 4-flow. Snarks are usually regarded to be, in some sense, the minimal cubic graphs without a 3-edge-colouring. We defined the notion of 4-flow-critical graphs as an alternative concept towards minimal graphs. It turns out that every snark has a 4-flow-critical snark as a minor. We verify, surprisingly, that less than 5% of the snarks with up to 28 vertices are 4-flow-critical. On the other hand, there are infinitely many 4-flow-critical snarks, as every flower-snark is 4-flow-critical. These observations give some insight into a new research approach regarding Tutteʼs Flow Conjectures.

We construct a pairing based simulation-extractable SNARK (SE-SNARK) that consists of only 3 group elements and has highly efficient verification. By formally linking SE-SNARKs to signatures of knowledge, we then obtain a succinct signature of knowledge consisting of only 3 group elements.

A snark is a cubic graph with no proper 3-edge-colouring. In 1996, Nedela and Skoviera proved the following theorem: Let G be a snark with an k -edge-cut, k ? 2 , whose removal leaves two 3-edge-colourable components M and N . Then both M and N can be completed to two snarks M ? and N ? of order not exceeding that of G by adding at most ? ( k ) vertices, where the number ? ( k ) only depends on k . The known values of the function ? ( k ) are ? ( 2 ) = 0 , ? ( 3 ) = 1 , ? ( 4 ) = 2 (Goldberg, 1981) 6], and ? ( 5 ) = 5 (Cameron et?al. 1987) 4]. The value ? ( 6 ) is not known and is apparently difficult to calculate. In 1979, Jaeger conjectured that there are no 7-cyclically-connected snarks. If this conjecture holds true, then ? ( 6 ) is the last important value to determine. The paper is aimed attacking the problem of determining ? ( 6 ) by investigating the structure and colour properties of potential complements in 6-decompositions of snarks. We find a set of 14 complements that suffice to perform 6 -decompositions of snarks with at most 30 vertices. We show that if this set is not complete to perform 6-decompositions of all snarks, then ? ( 6 ) ? 20 and there are strong restrictions on the structure of (possibly) missing complements. Part of the proofs are computer assisted.

In the paper we study $$\lambda $$ -numbers of several classes of snarks. We show that the $$\lambda $$ -number of each Blanu $$\breve{s}$$ a snark, Flower snark and Goldberg snark is $$6$$ . For $$n\ge 2$$ , we show that there is a dot product of $$n$$ Petersen graphs such that its $$\lambda $$ -number is 6.

For every even integer n ≥ 28 we construct a cyclically 4-edge-connected snark of order n which has girth 5 and contains a 5-circuit in every 2-factor. In addition, for every given positive integer k , we construct a nontrivial snark having at least k 5-circuits in every 2-factor.

Snarks are cubic bridgeless graphs of chromatic index 4 which had their origin in the search of counterexamples to the Four Color Theorem. 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. Since then, only two families of snarks have had their total chromatic number determined to be 4, namely the Flower Snark family and the Goldberg family. We prove that the total chromatic number of the Loupekhine family is 4. We study the dot product, a known operation to construct snarks. We consider families of snarks using the dot product, particularly subfamilies of the Blanusa families, and obtain a 4-total coloring for each family. We study edge coloring properties of girth trivial snarks that cannot be extended to total coloring. We classify the snark recognition problem as CoNP-complete and establish that the chromatic number of a snark is 3.

We show that for each rational number r such that 4

For each k, 1≤k≤n, we construct a dot product of n copies of the Petersen graph whose orientable genus is precisely k. We show that these are all possible values for the genus of Pn. This result gives counterexamples of all possible genera to a conjecture of Tinsley and Watkins from 1982. We show that the Petersen graph is the only Petersen power which can be embedded into the projective plane. For each k, 2≤k≤n − 1, we construct a Petersen power Pn whose non-orientable genus is precisely k. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:1-8, 2011

As I read Jennifer Jenkins's English as a Lingua Franca: Attitude and Identity, I thought of the Bellman who searches for the Snark, that fabulous creature, in Lewis Carroll's The Hunting of the Sn...

Flower snarks and Goldberg snarks are two infinite families of cyclically 5–edge–connected cubic graphs with girth at least five and chromatic index four. For any odd integer k, k > 3, there is a Flower snark, say Jk, of order 4k and a Goldberg snark, say Bk, of order 8k. We determine the automorphism groups of Jk and Bk for every k and prove that they are isomorphic to the dihedral group D4k of order 4k.