Special Classes of Snarks

We report the most relevant results on the classification, up to isomorphism, of nontrivial simple uncolorable (i.e., the chromatic index equals 4) cubic graphs, called snarks in the literature. Then we study many classes of snarks satisfying certain additional conditions, and investigate the relationships among them. Finally, we discuss connections between the snark family and some significant conjectures of graph theory, and list some problems and open questions which arise naturally in this research.     

Snarks and Flow-Critical Graphs

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.     

Snarks with given real flow numbers

We show that for each rational number r such that 4<r?5 there exist infinitely many cyclically 4‐edge‐connected cubic graphs of chromatic index 4 and girth at least 5—that is, snarks—whose flow number equals r. This answers a question posed by Pan and Zhu [Construction of graphs with given circular flow numbers, J Graph Theory 43 [2003], 304–318]. © 2011 Wiley Periodicals, Inc. J Graph Theory 68: 189‐201, 2011     

Infinite Classes of Dihedral Snarks

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 J _k , of order 4k and a Goldberg snark, say B _k , of order 8k. We determine the automorphism groups of J _k and B _k for every k and prove that they are isomorphic to the dihedral group D _4k of order 4k. Research performed within the activity of INdAM–GNSAGA with the financial support of the Italian Ministry MIUR, project “Strutture Geometriche, Combinatoria e loro Applicazioni”.     

Snarks,Hypohamiltonian Graphs and Non-Supereulerian Graphs

A graph G is hypohamiltonian if it is not Hamiltonian but for each \(v\in V(G)\), the graph \(G-v\) is Hamiltonian. A graph is supereulerian if it has a spanning Eulerian subgraph. A graph G is called collapsible if for every even subset \(R\subseteq V(G)\), there is a spanning connected subgraph H of G such that R is the set of vertices of odd degree in H. A graph is reduced if it has no nontrivial collapsible subgraphs. In this note, we first prove that all hypohamiltonian cubic graphs are reduced non-supereulerian graphs. Then we introduce an operation to construct graphs from hypohamiltonian cubic graphs such that the resulting graphs are 3-edge-connected non-supereulerian reduced graphs and cannot be contracted to a snark. This disproves two conjectures, one of which was first posed by Catlin et al. in [Congr. Num. 76:173–181, 1990] and in [J. Combin. Theory, Ser B 66:123–139, 1996], and was posed again by Li et al. in [Acta Math. Sin. English Ser 30(2):291–304, 2014] and by Yang in [Supereulerian graphs, hamiltonicity of graphs and several extremal problems in graphs, Ph. D. Dissertation, Université Paris-Sub, September 27, 2013], respectively, the other one was posed by Yang 2013.     

Cayley Snarks and Almost Simple Groups

A Cayley snark is a cubic Cayley graph which is not 3-edge-colourable. In the paper we discuss the problem of the existence of Cayley snarks. This problem is closely related to the problem of the existence of non-hamiltonian Cayley graphs and to the question whether every Cayley graph admits a nowhere-zero 4-flow. So far, no Cayley snarks have been found. On the other hand, we prove that the smallest example of a Cayley snark, if it exists, comes either from a non-abelian simple group or from a group which has a single non-trivial proper normal subgroup. The subgroup must have index two and must be either non-abelian simple or the direct product of two isomorphic non-abelian simple groups. Received January 18, 2000 Research partially supported by VEGA grant 1/3213/96 Research partially supported by VEGA grants 1/3213/96 and 1/4318/97     

Uniquely Edge-3-Colorable Graphs and Snarks

 A cubic graph G is uniquely edge-3-colorable if G has precisely one 1-factorization. It is proved in this paper, if a uniquely edge-3-colorable, cubic graph G is cyclically 4-edge-connected, but not cyclically 5-edge-connected, then G must contain a snark as a minor. This is an approach to a conjecture that every triangle free uniquely edge-3-colorable cubic graph must have the Petersen graph as a minor. Fiorini and Wilson (1976) conjectured that every uniquely edge-3-colorable planar cubic graph must have a triangle. It is proved in this paper that every counterexample to this conjecture is cyclically 5-edge-connected and that in a minimal counterexample to the conjecture, every cyclic 5-edge-cut is trivial (an edge-cut T of G is cyclic if no component of G\T is acyclic and a cyclic edge-cut T is trivial if one component of G\T is a circuit of length |T|). Received: July 14, 1997 Revised: June 11, 1998     

Computing the Automorphic Chromatic Index of Certain Snarks

The automorphic G-chromatic index of a graph Γ is the minimum integer m for which Γ has a proper edge-coloring with m colors which is preserved by the full automorphism group G of Γ. We determine the automorphic G-chromatic index of each member of four infinite classes of snarks: type I Blanu?a snarks, type II Blanu?a snarks, Flower snarks and Goldberg snarks.     

Generation of Cubic Graphs and Snarks with Large Girth

We describe two new algorithms for the generation of all non‐isomorphic cubic graphs with girth at least that are very efficient for 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.     

Families of Dot-Product Snarks on Orientable Surfaces of Low Genus

We introduce a generalized dot product and provide some embedding conditions under which the genus of a graph does not rise under a dot product with the Petersen graph. Using these conditions, we disprove a conjecture of Tinsley and Watkins on the genus of dot products of the Petersen graph and show that both Grünbaum’s Conjecture and the Berge-Fulkerson Conjecture hold for certain infinite families of snarks. Additionally, we determine the orientable genus of four known snarks and two known snark families, construct a new example of an infinite family of snarks on the torus, and construct ten new examples of infinite families of snarks on the 2-holed torus; these last constructions allow us to show that there are genus-2 snarks of every even order n ≥ 18.     

Contractible Subgraphs,Thomassen's Conjecture and the Dominating Cycle Conjecture for Snarks

We show that the conjectures by Matthews and Sumner (every 4-connected claw-free graph is hamiltonian), by Thomassen (every 4-connected line graph is hamiltonian) and by Fleischner (every cyclically 4-edge-connected cubic graph has either a 3-edge-coloring or a dominating cycle), which are known to be equivalent, are equivalent with the statement that every snark (i.e. a cyclically 4-edge-connected cubic graph of girth at least five that is not 3-edge-colorable) has a dominating cycle.We use a refinement of the contractibility technique which was introduced by Ryjáček and Schelp in 2003 as a common generalization and strengthening of the reduction techniques by Catlin and Veldman and of the closure concept introduced by Ryjáček in 1997.     

Critical and Critical Tangent Cones in Optimization Problems

In this paper the notion of critical tangent cone CT(x|Q) to Q at x is introduced for the case when Q is a convex subset of a normed space X. If Q is closed with nonempty interior, and x∈Q, the nonemptiness of the Dubovitskii–Milyutin set of second-order admissible variations, V(x,d|Q), is then characterized by the condition d∈CT(x|Q). Furthermore, the support function of V(x,d|Q) is shown to be evaluated in terms of that support function of Q. For the cases when Q is the set of continuous or L ^∞ selections of a certain set-valued map, the corresponding characterization of the cone CT(x|Q) and the formula for the support function of V(x,d|Q) are obtained in terms of more verifiable conditions.     

Stability of Critical Values and Isolated Critical Continua

We extend the study of critical points in [4]. We show that isolated components of critical points lying on a levelset can be described by an integer which is a lower bound to the “number” of critical points of any function near to the original one in C¹-sup-norm. We also derive a global theorem about continua of critical values similar to that given by Rabinowitz for continua of solutions of certain nonlinear eigenvalue problems. We give a simple application of our abstract results to the problem of bifurcation for gradient systems when the linearization is not completely continuous.     

Critical theories

Sverdlovsk. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 29, No. 1, pp. 23–31, January–February, 1988.     

Critical duality

We look for a general framework in which the Ekeland duality can be formulated. We propose a scheme in which the parameter sets are provided with a coupling function which induces a conjugacy. The decision spaces are not supposed to have any special structure. We examine several examples. In particular, we consider some special classes of generalized convex functions.     

On Critical Values of Polynomials with Real Critical Points

Let f be a polynomial of degree at least 2 with f(0)=0 and f′(0)=1. Suppose that all the zeros of f′ are real. We show that there is a zero ζ of f′ such that |f(ζ)/ζ|≤2/3, and that this inequality can be taken to be strict unless f is of the form f(z)=z+cz ³.