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Regular polygonal systems
Jurij Kovič, 2019, izvirni znanstveni članek

Ključne besede: regular polygonal system, boundary code, face vector, symmetry group, reconstructibility from the boundary
Objavljeno v RUP: 03.01.2022; Ogledov: 379; Prenosov: 17
.pdf Polno besedilo (353,82 KB)

Splittable and unsplittable graphs and configurations
Nino Bašić, Jan Grošelj, Branko Grünbaum, Tomaž Pisanski, 2019, izvirni znanstveni članek

Opis: We prove that there exist infinitely many splittable and also infinitely many unsplittable cyclic ▫$(n_3)$▫ configurations. We also present a complete study of trivalent cyclic Haar graphs on at most 60 vertices with respect to splittability. Finally, we show that all cyclic flag-transitive configurations with the exception of the Fano plane and the Möbius-Kantor configuration are splittable.
Ključne besede: configuration of points and lines, unsplittable configuration, unsplittable graph, independent set, Levi graph, Grünbaum graph, splitting type, cyclic Haar graph
Objavljeno v RUP: 03.01.2022; Ogledov: 386; Prenosov: 19
.pdf Polno besedilo (355,79 KB)

Semiregular automorphisms in vertex-transitive graphs with a solvable group of automorphisms
Dragan Marušič, 2017, izvirni znanstveni članek

Opis: It has been conjectured that automorphism groups of vertex-transitive (di)graphs, and more generally 2-closures of transitive permutation groups, must necessarily possess a fixed-point-free element of prime order, and thus a non-identity element with all orbits of the same length, in other words, a semiregular element. The known affirmative answers for graphs with primitive and quasiprimitive groups of automorphisms suggest that solvable groups need to be considered if one is to hope for a complete solution of this conjecture. It is the purpose of this paper to present an overview of known results and suggest possible further lines of research towards a complete solution of the problem.
Ključne besede: solvable group, semiregular automorphism, fixed-point-free automorphism, polycirculant conjecture
Objavljeno v RUP: 03.01.2022; Ogledov: 413; Prenosov: 17
.pdf Polno besedilo (235,26 KB)

A note on acyclic number of planar graphs
Mirko Petruševski, Riste Škrekovski, 2017, izvirni znanstveni članek

Opis: The acyclic number ▫$a(G)$▫ of a graph ▫$G$▫ is the maximum order of an induced forest in ▫$G$▫. The purpose of this short paper is to propose a conjecture that ▫$a(G)\geq \left( 1-\frac{3}{2g}\right)n$▫ holds for every planar graph ▫$G$▫ of girth ▫$g$▫ and order ▫$n$▫, which captures three known conjectures on the topic. In support of this conjecture, we prove a weaker result that ▫$a(G)\geq \left( 1-\frac{3}{g} \right)n$▫ holds. In addition, we give a construction showing that the constant ▫$\frac{3}{2}$▫ from the conjecture cannot be decreased.
Ključne besede: induced forest, acyclic number, planar graph, girth
Objavljeno v RUP: 03.01.2022; Ogledov: 435; Prenosov: 15
.pdf Polno besedilo (227,50 KB)

Classification of convex polyhedra by their rotational orbit Euler characteristic
Jurij Kovič, 2017, izvirni znanstveni članek

Opis: Let ▫$\mathcal P$▫ be a polyhedron whose boundary consists of flat polygonal faces on some compact surface ▫$S(\mathcal P)$▫ (not necessarily homeomorphic to the sphere ▫$S^{2}$)▫. Let ▫$vo_{R}(\mathcal P), eo_{R}(\mathcal P)$▫, ▫$ fo_{R}(\mathcal P)$▫ be the numbers of rotational orbits of vertices, edges and faces, respectively, determined by the group ▫$G = G_{R}(P)$▫ of all the rotations of the Euclidean space ▫$E^{3}$▫ preserving ▫$\mathcal P$▫. We define the ''rotational orbit Euler characteristic'' of ▫$\mathcal P$▫ as the number ▫$Eo_{R}(\mathcal P) = vo_{R}(\mathcal P) - eo_{R}(\mathcal P) + fo_{R}(\mathcal P)$▫. Using the Burnside lemma we obtain the lower and the upper bound for ▫$Eo_{R}(\mathcal P)$▫ in terms of the genus of the surface ▫$S(P)$▫. We prove that ▫$Eo_{R} \in \lbrace 2,1,0,-1\rbrace $▫ for any convex polyhedron ▫$\mathcal P$▫. In the non-convex case ▫$Eo_{R}$▫ may be arbitrarily large or small.
Ključne besede: polyhedron, rotational orbit, Euler characteristic
Objavljeno v RUP: 03.01.2022; Ogledov: 353; Prenosov: 16
.pdf Polno besedilo (272,96 KB)

Vertex-transitive graphs and their arc-types
Marston D. E. Conder, Tomaž Pisanski, Arjana Žitnik, 2017, izvirni znanstveni članek

Opis: Let ▫$X$▫ be a finite vertex-transitive graph of valency ▫$d$▫, and let ▫$A$▫ be the full automorphism group of ▫$X$▫. Then the arc-type of ▫$X$▫ is defined in terms of the sizes of the orbits of the stabiliser ▫$A_v$▫ of a given vertex ▫$v$▫ on the set of arcs incident with ▫$v$▫. Such an orbit is said to be self-paired if it is contained in an orbit ▫$\Delta$▫ of ▫$A$▫ on the set of all arcs of v$X$▫ such that v$\Delta$▫ is closed under arc-reversal. The arc-type of ▫$X$▫ is then the partition of ▫$d$▫ as the sum ▫$n_1 + n_2 + \dots + n_t + (m_1 + m_1) + (m_2 + m_2) + \dots + (m_s + m_s)$▫, where ▫$n_1, n_2, \dots, n_t$▫ are the sizes of the self-paired orbits, and ▫$m_1,m_1, m_2,m_2, \dots, m_s,m_s$▫ are the sizes of the non-self-paired orbits, in descending order. In this paper, we find the arc-types of several families of graphs. Also we show that the arc-type of a Cartesian product of two "relatively prime" graphs is the natural sum of their arc-types. Then using these observations, we show that with the exception of ▫$1+1$▫ and ▫$(1+1)$▫, every partition as defined above is \emph{realisable}, in the sense that there exists at least one vertex-transitive graph with the given partition as its arc-type.
Ključne besede: symmetry type, vertex-transitive graph, arc-transitive graph, Cayley graph, cartesian product, covering graph
Objavljeno v RUP: 03.01.2022; Ogledov: 382; Prenosov: 16
.pdf Polno besedilo (475,17 KB)

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