Ján Karabáš

Slovak University of Technology in Bratislava · Institute of Computer Science and Mathematics

The colouring defect of a cubic graph is the smallest number of edges left uncovered by any set of three perfect matchings. While $3$-edge-colourable graphs have defect $0$, those that cannot be $3$-edge-coloured (that is, snarks) are known to have defect at least $3$. In this paper we focus on the structure and properties of snarks with defect $3$...

The colouring defect of a cubic graph is the smallest number of edges left uncovered by any set of three perfect matchings. While $3$-edge-colourable graphs have defect $0$, those that cannot be $3$-edge-coloured (that is, snarks) are known to have defect at least $3$. In this paper we focus on the structure and properties of snarks with defect $3$...

The colouring defect of a cubic graph is the smallest number of edges left uncovered by any set of three perfect matchings. While $3$-edge-colourable graphs have defect~$0$, those that cannot be $3$-edge-coloured have defect at least $3$. We show that every bridgeless cubic graph with defect $3$ can have its edges covered with at most five perfect...

The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect 0 if and only if it is 3-edge-colourable, this invariant can measure how much a cubic graph differs from a 3-edge-colourable graph. Our aim is to examine t...

A long-standing conjecture of Berge suggests that every bridgeless cubic graph can be expressed as a union of at most five perfect matchings. This conjecture trivially holds for $3$-edge-colourable cubic graphs, but remains widely open for graphs that are not $3$-edge-colourable. The aim of this paper is to verify the validity of Berge's conjecture...

In the present paper we investigate the faithfulness of certain linear representations of groups of automorphisms of a graph $X$ in the group of symmetries of the Jacobian of $X$. As a consequence we show that if $X$ is not a cycle and it admits a non-abelian semiregular group of automorphims, then the Jacobian of $X$ cannot be cyclic. While the si...

In 2006 M. Conder published the list of orientably regular hypermaps up to genus $101$ as well as the list of large groups acting on orientable surfaces up to genus $101$. The regular hypermaps are closely related to actions of triangle groups on closed orientable surfaces, whereas the large groups are groups acting with either triangular or quadra...

We investigate a representation of the automorphism group of a connected graph X in the group of unimodular matrices Uβ of dimension β, where β is the Betti number of graph X. We classify the graphs for which the automorphism group does not embed into Uβ. It follows that if X has no pendant vertices and X is not a simple cycle, then the representat...

The colouring defect of a cubic graph, introduced by Steffen in 2015, is the minimum number of edges that are left uncovered by any set of three perfect matchings. Since a cubic graph has defect 0 if and only if it is 3-edge-colourable, this invariant can measure how much a cubic graph differs from a 3-edge-colourable graph. Our aim is to examine t...

We investigate a representation of the automorphism group of a connected graph X in the group of unimodular matrices Uβ of dimension β, where β is the Betti number of graph X. We classify the graphs for which the automorphism group does not embed into Uβ. It follows that if X has no pendant vertices and X is not a simple cycle, then the representat...

In this paper we generalise Singerman’s results on triangle group inclusions to the broader class of generalised quadrangle groups, that is, Fuchsian groups with signature of genus $0$ and generated by three or four elliptic generators. For any possible inclusion $P<Q$ we also give the number of non-conjugate subgroups of $Q$ isomorphic to $P$.

The well known infinite families of prisms and antiprisms on the sphere were for long time not considered as Archimedean solids for reasons not fully understood. In this paper we describe the first two infinite families of Archimedean maps on higher genera which we call " generalised " prisms and " generalised " antiprisms.

In a classical result of 1972 Singerman classifies the inclusions between triangle groups. We extend the classification to a broader family of triangle and quadrangle groups forming a particular subfamily of Fuchsian groups. With two exceptions, each inclusion determines a finite bipartite map (hypermap) on a 2-dimensional spherical orbifold, calle...

In the present paper we introduce a family of functors (called operations) of the category of hypermaps (dessins) preserving the underlying Riemann surface. The considered family of functors include as particular instances the operations considered by Magot and Zvonkin (2000), Singerman and Syddall (2003), and Girondo (2003). We identify a set of 1...

Discrete actions of finite groups on surfaces appears in many situations in numerous branches of mathematics, cryptography, quantum physics, and many other fields of science. In topological graph theory they can be used to derive lists of highly symmetrical maps of fixed genus: regular maps, vertex-transitive maps, Cayley maps, or edge-transitive m...

The result is of crucial importance in map and graph enumeration problems. The procedure for obtaining the result is quite simple. The classification procedure is based on a result of Harvey: ”The maximal order n of a cyclic group acting over an orientable surface is bounded; n<=4g + 2, where g is genus of the surface”. Another important fact can b...

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(k) vertices,...

The paper focuses on the classification of vertex-transitive polyhedral maps of genus from 2 to 4. These maps naturally generalise the spherical maps associated with the classical Archimedean solids. Our analysis is based on the fact that each Archimedean map on an orientable surface projects onto a one- or a two-vertex quotient map. For a given ge...

Here is the complete census of Archimedean maps of genera from two to four. Presented data are based on the newest revision my paper "Archimedean solids of higher genera" which I wrote with Roman Nedela. BCK classification is the listing of classes of Archimedean maps with regard to the paper "Two infinite families of Archimedean maps of higher gen...

It is known that every closed compact orientable 3-manifold M can be represented by a 4-edge-coloured 4-valent graph called a crystallisation of M. Casali and Grasselli proved that 3-manifolds of Heegaard genus g can be represented by crystallisations with a very simple structure which can be described by a 2(g+1)-tuple of non-negative integers. Th...

The problem of classifying orientable vertex-transitive maps on a surface with genus two is considered. We construct and classify all simple orientable vertex-transitive maps, with face width at least 3 which can be viewed as generalisations of classical Archimedean solids. The proof is computer-aided. The developed method applies to higher genera...

Present paper deals with fundamental groups of 3-manifolds represented by certain family S of bipartite 4-edge-coloured graphs. List of fundamental goups of prime 3-manifolds with genus two represented by graphs in S with at most 42 vertices is produced.

One of the central problems for 3-manifolds is the isomorphism problem. Since 70's several methods to attack it were developed. The method introduced in a paper of Ferri and Gagliardi is not easy to use, since no bound for the number of steps in a computer representation is known. Some approximations were introduced in the paper of Grasselli, Mulaz...