1. Hamiltonicity of vertex-transitive graphs of order 4pKlavdija Kutnar, Dragan Marušič, 2008, izvirni znanstveni članek Opis: It is shown that every connected vertex-transitive graph of order ▫$4p$▫, where ▫$p$▫ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path. Najdeno v: ključnih besedah Povzetek najdenega: ...graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group... Ključne besede: graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group Objavljeno: 15.10.2013; Ogledov: 1518; Prenosov: 17 Polno besedilo (0,00 KB) |
2. On quartic half-arc-transitive metacirculantsDragan Marušič, Primož Šparl, 2008, izvirni znanstveni članek Opis: 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. Najdeno v: ključnih besedah Povzetek najdenega: ...graph is a graph admitting a transitive group generated by two automorphisms ▫$\rho$▫ and ▫$\sigma$▫,... ...mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism... Ključne besede: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group Objavljeno: 15.10.2013; Ogledov: 1574; Prenosov: 44 Polno besedilo (0,00 KB) |
3. Distance-regular Cayley graphs on dihedral groupsŠtefko Miklavič, Primož Potočnik, 2007, izvirni znanstveni članek Opis: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices. Najdeno v: ključnih besedah Povzetek najdenega: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such... ...mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral... Ključne besede: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set Objavljeno: 15.10.2013; Ogledov: 1173; Prenosov: 41 Polno besedilo (0,00 KB) |
4. On 2-fold covers of graphsYan-Quan Feng, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2008, izvirni znanstveni članek Opis: A regular covering projection ▫$\wp : \widetilde{X} \to X$▫ of connected graphs is ▫$G$▫-admissible if ▫$G$▫ lifts along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of covering transformations. The projection is called ▫$G$▫-split whenever the extension ▫{$\mathrm{CT}}(\wp) \to \tilde{G} \to G$▫ splits. In this paper, split 2-covers are considered, with a particular emphasis given to cubic symmetric graphs. Supposing that ▫$G$▫ is transitive on ▫$X$▫, a ▫$G$▫-split cover is said to be ▫$G$▫-split-transitive if all complements ▫$\tilde{G} \cong G$▫ of CT▫$(\wp)$▫ within ▫$\tilde{G}$▫ are transitive on ▫$\widetilde{X}$▫; it is said to be ▫$G$▫-split-sectional whenever for each complement ▫$\tilde{G}$▫ there exists a ▫$\tilde{G}$▫-invariant section of ▫$\wp$▫; and it is called ▫$G$▫-split-mixed otherwise. It is shown, when ▫$G$▫ is an arc-transitive group, split-sectional and split-mixed 2-covers lead to canonical double covers. Split-transitive covers, however, are considerably more difficult to analyze. For cubic symmetric graphs split 2-cover are necessarily canonical double covers (that is, no ▫$G$▫-split-transitive 2-covers exist) when ▫$G$▫ is 1-regular or 4-regular. In all other cases, that is, if ▫$G$▫ is ▫$s$▫-regular, ▫$s=2,3$▫ or ▫$5$▫, a necessary and sufficient condition for the existence of a transitive complement ▫$\tilde{G}$▫ is given, and moreover, an infinite family of split-transitive 2-covers based on the alternating groups of the form ▫$A_{12k+10}$▫ is constructed. Finally, chains of consecutive 2-covers, along which an arc-transitive group ▫$G$▫ has successive lifts, are also considered. It is proved that in such a chain, at most two projections can be split. Further, it is shown that, in the context of cubic symmetric graphs, if exactly two of them are split, then one is split-transitive and the other one is either split-sectional or split-mixed. Najdeno v: ključnih besedah Povzetek najdenega: ...along ▫$\wp$▫. Denote by ▫$\tilde{G}$▫ the lifted group, and let CT▫$(\wp)$▫ be the group of... ...graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group,... Ključne besede: graph theory, graphs, cubic graphs, symmetric graphs, ▫$s$▫-regular group, regular covering projection Objavljeno: 15.10.2013; Ogledov: 1166; Prenosov: 13 Polno besedilo (0,00 KB) |
5. Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power orderEdward Dobson, 2010, izvirni znanstveni članek Opis: We show that almost every Cayley graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order has automorphism group as small as possible. Additionally, we show that almost every Cayley (di)graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order that does not have automorphism group as small as possible is a normal Cayley (di)graph of ▫$G$▫ (that is, ▫$G_L \triangleleft {\rm Aut}(\Gamma))$▫. Najdeno v: ključnih besedah Povzetek najdenega: ...every Cayley graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of odd prime-power order has automorphism... ...mathematics, graph theory, Cayley graph, abelian group, automorphism group, asymptotic,... Ključne besede: mathematics, graph theory, Cayley graph, abelian group, automorphism group, asymptotic, ▫$p$▫-group Objavljeno: 15.10.2013; Ogledov: 2346; Prenosov: 39 Polno besedilo (0,00 KB) |
6. Hamilton paths and cycles in vertex-transitive graphs of order 6pKlavdija Kutnar, Primož Šparl, 2009, izvirni znanstveni članek Opis: 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. Najdeno v: ključnih besedah Povzetek najdenega: ...graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group... Ključne besede: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Objavljeno: 15.10.2013; Ogledov: 1580; Prenosov: 14 Polno besedilo (0,00 KB) |
7. Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groupsŠtefko Miklavič, Primož Šparl, 2012, izvirni znanstveni članek Opis: 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. Najdeno v: ključnih besedah Povzetek najdenega: ...class of connected Cayley graphs on abelian groups. It is proved that every connected Cayley... ...graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable,... Ključne besede: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian group Objavljeno: 15.10.2013; Ogledov: 1179; Prenosov: 55 Polno besedilo (0,00 KB) |
8. Arc-transitive cycle decompositions of tetravalent graphsŠtefko Miklavič, Primož Potočnik, Stephen Wilson, 2008, izvirni znanstveni članek Opis: A cycle decomposition of a graph ▫$\Gamma$▫ is a set ▫$\mathcal{C}$▫ of cycles of ▫$\Gamma$▫ such that every edge of ▫$\Gamma$▫ belongs to exactly one cycle in ▫$\mathcal{C}$▫. Such a decomposition is called arc-transitive if the group of automorphisms of ▫$\Gamma$▫ that preserve setwise acts transitively on the arcs of ▫$\Gamma$▫. In this paper, we study arc-transitive cycle decompositions of tetravalent graphs. In particular, we are interested in determining and enumerating arc-transitive cycle decompositions admitted by a given arc-transitive tetravalent graph. Among other results we show that a connected tetravalent arc-transitive graph is either 2-arc-transitive, or is isomorphic to the medial graph of a reflexible map, or admits exactly one cycle structure. Najdeno v: ključnih besedah Povzetek najdenega: ...a decomposition is called arc-transitive if the group of automorphisms of ▫$\Gamma$▫ that preserve setwise... ...mathematics, graph theory, cycle decomposition, automorphism group, consistent cycle, medial... Ključne besede: mathematics, graph theory, cycle decomposition, automorphism group, consistent cycle, medial maps Objavljeno: 15.10.2013; Ogledov: 1306; Prenosov: 33 Polno besedilo (0,00 KB) |
9. On overgroups of regular abelian p-groupsEdward Dobson, 2009, izvirni znanstveni članek Opis: Let ▫$G$▫ be a transitive group of odd prime-power degree whose Sylow ▫$p$▫-subgroup ▫$P$▫ is abelian od rank ▫$t$▫. Weshow that if ▫$p > 2^{t-1}$▫, then ▫$G$▫ has a normal subgroup that is a direct product of ▫$t$▫ permutation groups of smaller degree that are either cyclic or doubly-transitive simple groups. As a consequence, we determine the full automorphism group of a Cayley diagraph of an abelian group with rank two such that the Sylow ▫$p$▫-subgroup of the full automorphism group is abelian. Najdeno v: ključnih besedah Povzetek najdenega: ...Let ▫$G$▫ be a transitive group of odd prime-power degree whose Sylow ▫$p$▫-subgroup... ...group theory, graph theory, Cayley graph, abelian group, regular... Ključne besede: group theory, graph theory, Cayley graph, abelian group, regular group, p-group Objavljeno: 15.10.2013; Ogledov: 1485; Prenosov: 42 Polno besedilo (0,00 KB) |
10. Distance-regular Cayley graphs on dihedral groupsPrimož Potočnik, Štefko Miklavič, 2005, izvirni znanstveni članek Opis: The main result of this article is a classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such graphs, which are called trivial. These are: complete graphs, complete bipartite graphs, complete bipartite graphs with the edges of a 1-factor removed, and cycles. It is proved that every non-trivial distance-regular Cayley graph on a dihedral group is bipartite, non-antipodal, has diameter 3 and arises either from a cyclic di#erence set, or possibly (if any such exists) from a dihedral difference set satisfying some additional conditions. Finally, all distance-transitive Cayley graphs on dihedral groups are determined. It transpires that a Cayley graph on a dihedral group is distance-transitive if and only if it is trivial, or isomorphic to the incidence or to the non-incidence graph of a projective space ▫$\mathrm{PG}_{d-1} (d,q)$▫, ▫$d \ge 2$▫, or the unique pair of complementary symmetric designs on 11 vertices. Najdeno v: ključnih besedah Povzetek najdenega: ...classification of distance-regular Cayley graphs on dihedral groups. There exist four obvious families of such... ...mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral... Ključne besede: mathematics, grah theory, distance-regular graph, distance-transitive graph, Cayley graph, dihedral group, dihedrant, difference set Objavljeno: 10.07.2015; Ogledov: 798; Prenosov: 34 Polno besedilo (0,00 KB) |