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 1 - 10 / 29123 1.Graphs and their automorphism groups bpredavanje eUnidad Cuernavaca del Instituto de Matemáticas, UNAM, Cuernavaca, Morelos, Mehika, 23. - 27. 7. 2012Klavdija Kutnar, Primož Šparl, 2012, invited lecture at foreign universityFound in: osebiKeywords: automorphism groupPublished: 15.10.2013; Views: 1813; Downloads: 11 Full text (0,00 KB) 2.On quartic half-arc-transitive metacirculantsDragan Marušič, Primož Šparl, 2008, original scientific articleAbstract: Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫, where ▫$\rho$▫ is ▫$(m,n)$▫-semiregular for some integers ▫$m \ge 1$▫, ▫$n \ge 2▫$, and where ▫$\sigma$▫ normalizes ▫$\rho$▫, cyclically permuting the orbits of ▫$\rho$▫ in such a way that ▫$\sigma^m$▫ has at least one fixed vertex. A half-arc-transitive graph is a vertex- and edge- but not arc-transitive graph. In this article quartic half-arc-transitive metacirculants are explored and their connection to the so called tightly attached quartic half-arc-transitive graphs is explored. It is shown that there are three essentially different possibilities for a quartic half-arc-transitive metacirculant which is not tightly attached to exist. These graphs are extensively studied and some infinite families of such graphs are constructed.Found in: osebiKeywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism groupPublished: 15.10.2013; Views: 1898; Downloads: 78 Full text (0,00 KB) 3.On the connectivity of bipartite distance-balanced graphsŠtefko Miklavič, Primož Šparl, 2012, original scientific articleAbstract: A connected graph ▫$\varGamma$▫ is said to be distance-balanced whenever for any pair of adjacent vertices ▫$u,v$▫ of ▫$\varGamma$▫ the number of vertices closer to ▫$u$▫ than to ▫$v$▫ is equal to the number of vertices closer to ▫$v$▫ than to ▫$u$▫. In [K. Handa, Bipartite graphs with balanced ▫$(a,b)$▫-partitions, Ars Combin. 51 (1999), 113-119] Handa asked whether every bipartite distance-balanced graph, that is not a cycle, is 3-connected. In this paper the Handa question is answered in the negative. Moreover, we show that a minimal bipartite distance-balanced graph, that is not a cycle and is not 3-connected, has 18 vertices and is unique. In addition, we give a complete classification of non-3-connected bipartite distance-balanced graphs for which the minimal distance between two vertices in a 2-cut is three. All such graphs are regular and for each ▫$k \geq 3$▫ there exists an infinite family of such graphs which are ▫$k$▫-regular.Furthermore, we determine a number of structural properties that a bipartite distance-balanced graph, which is not 3-connected, must have. As an application, we give a positive answer to the Handa question for the subfamily of bipartite strongly distance-balanced graphs.Found in: osebiKeywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphsPublished: 15.10.2013; Views: 1576; Downloads: 57 Full text (0,00 KB) 4.Strongly regular tri-Cayley graphsKlavdija Kutnar, Dragan Marušič, Štefko Miklavič, Primož Šparl, 2009, original scientific articleAbstract: A graph is called tri-Cayley if it admits a semiregular subgroup of automorphisms having three orbits of equal length. In this paper, the structure of strongly regular tri-Cayley graphs is investigated. A structural description of strongly regular tri-Cayley graphs of cyclic groups is given.Found in: osebiKeywords: strongly regular graph, tri-Cayley graphPublished: 15.10.2013; Views: 1469; Downloads: 59 Full text (0,00 KB) 5.Hamilton paths and cycles in vertex-transitive graphs of order 6pKlavdija Kutnar, Primož Šparl, 2009, original scientific articleAbstract: It is shown that every connected vertex-transitive graph of order ▫$6p$▫, where ▫$p$▫ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order ▫$6p$▫ which is not genuinely imprimitive contains a Hamilton cycle.Found in: osebiKeywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism groupPublished: 15.10.2013; Views: 1913; Downloads: 15 Full text (0,00 KB) 6.On Hamiltonicity of circulant digraphs of outdegree threeŠtefko Miklavič, Primož Šparl, 2009, original scientific articleAbstract: This paper deals with Hamiltonicity of connected loopless circulant digraphs of outdegree three with connection set of the form ▫$\{a,ka,c\}$▫, where ▫$k$▫ is an integer. In particular, we prove that if ▫$k=-1$▫ or ▫$k=2$▫ such a circulant digraph is Hamiltonian if and only if it is not isomorphic to the circulant digraph on 12 vertices with connection set ▫$\{3,6,4\}$▫.Found in: osebiKeywords: graph theory, circulant digraph, Hamilton cyclePublished: 15.10.2013; Views: 1475; Downloads: 59 Full text (0,00 KB) 7.Classification of half-arc-transitive graphs of order 4pKlavdija Kutnar, Dragan Marušič, Primož Šparl, Ru-Ji Wang, Ming-Yao Xu, 2013, original scientific articleFound in: osebiKeywords: graphPublished: 15.10.2013; Views: 1287; Downloads: 10 Full text (0,00 KB) 8.Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groupsŠtefko Miklavič, Primož Šparl, 2012, original scientific articleAbstract: In this paper the concepts of Hamilton cycle (HC) and Hamilton path (HP) extendability are introduced. A connected graph ▫$\Gamma$▫ is ▫$n$▫-HC-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton cycle of ▫$\Gamma$▫. Similarly, ▫$\Gamma$▫ is weakly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if every such path is contained in some Hamilton path of ▫$\Gamma$▫. Moreover, ▫$\Gamma$▫ is strongly ▫$n$▫-HP-extendable if it contains a path of length ▫$n$▫ and if for every such path $P$ there is a Hamilton path of ▫$\Gamma$▫ starting with ▫$P$▫. These concepts are then studied for the class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley graph on an abelian group of order at least three is 2-HC-extendable and a complete classification of 3-HC-extendable connected Cayley graphs of abelian groups is obtained. Moreover, it is proved that every connected Cayley graph on an abelian group of order at least five is weakly 4-HP-extendable.Found in: osebiKeywords: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian groupPublished: 15.10.2013; Views: 1424; Downloads: 86 Full text (0,00 KB) 9.Bled'11Klavdija Kutnar, Primož Šparl, 2013, preface, afterwordFound in: osebiPublished: 15.10.2013; Views: 1131; Downloads: 25 Full text (0,00 KB) 10.Hamiltonian cycles in Cayley graphs whose order has few prime factorsKlavdija Kutnar, Dragan Marušič, D. W. Morris, Joy Morris, Primož Šparl, 2012, original scientific articleAbstract: We prove that if Cay▫$(G; S)$▫ is a connected Cayley graph with ▫$n$▫ vertices, and the prime factorization of ▫$n$▫ is very small, then Cay▫$(G; S)$▫ has a hamiltonian cycle. More precisely, if ▫$p$▫, ▫$q$▫, and ▫$r$▫ are distinct primes, then ▫$n$▫ can be of the form kp with ▫$24 \ne k < 32$▫, or of the form ▫$kpq$▫ with ▫$k \le 5$▫, or of the form ▫$pqr$▫, or of the form ▫$kp^2$▫ with ▫$k \le 4$▫, or of the form ▫$kp^3$▫ with ▫$k \le 2$▫.Found in: osebiKeywords: graph theory, Cayley graphs, hamiltonian cyclesPublished: 15.10.2013; Views: 1819; Downloads: 74 Full text (0,00 KB)
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