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1.
2.
On quartic half-arc-transitive metacirculants
Dragan Marušič, Primož Šparl, 2008, original scientific article

Abstract: 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: osebi
Keywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group
Published: 15.10.2013; Views: 1635; Downloads: 57
URL Full text (0,00 KB)

3.
On the connectivity of bipartite distance-balanced graphs
Štefko Miklavič, Primož Šparl, 2012, original scientific article

Abstract: 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: osebi
Keywords: graph theory, connected graphs, connectivity, distance-balanced graphs, bipartite graphs
Published: 15.10.2013; Views: 1308; Downloads: 47
URL Full text (0,00 KB)

4.
Strongly regular tri-Cayley graphs
Klavdija Kutnar, Dragan Marušič, Štefko Miklavič, Primož Šparl, 2009, original scientific article

Abstract: 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: osebi
Keywords: strongly regular graph, tri-Cayley graph
Published: 15.10.2013; Views: 1230; Downloads: 47
URL Full text (0,00 KB)

5.
Hamilton paths and cycles in vertex-transitive graphs of order 6p
Klavdija Kutnar, Primož Šparl, 2009, original scientific article

Abstract: 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: osebi
Keywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group
Published: 15.10.2013; Views: 1629; Downloads: 14
URL Full text (0,00 KB)

6.
On Hamiltonicity of circulant digraphs of outdegree three
Štefko Miklavič, Primož Šparl, 2009, original scientific article

Abstract: 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: osebi
Keywords: graph theory, circulant digraph, Hamilton cycle
Published: 15.10.2013; Views: 1274; Downloads: 46
URL Full text (0,00 KB)

7.
Classification of half-arc-transitive graphs of order 4p
Klavdija Kutnar, Dragan Marušič, Primož Šparl, Ru-Ji Wang, Ming-Yao Xu, 2013, original scientific article

Found in: osebi
Keywords: graph
Published: 15.10.2013; Views: 1161; Downloads: 10
URL Full text (0,00 KB)

8.
Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groups
Štefko Miklavič, Primož Šparl, 2012, original scientific article

Abstract: 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: osebi
Keywords: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian group
Published: 15.10.2013; Views: 1226; Downloads: 70
URL Full text (0,00 KB)

9.
Bled'11
Klavdija Kutnar, Primož Šparl, 2013, preface, afterword

Found in: osebi
Published: 15.10.2013; Views: 1013; Downloads: 22
URL Full text (0,00 KB)

10.
Hamiltonian cycles in Cayley graphs whose order has few prime factors
Klavdija Kutnar, Dragan Marušič, D. W. Morris, Joy Morris, Primož Šparl, 2012, original scientific article

Abstract: 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: osebi
Keywords: graph theory, Cayley graphs, hamiltonian cycles
Published: 15.10.2013; Views: 1566; Downloads: 60
URL Full text (0,00 KB)

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