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2. 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. Ključne besede: graph theory, Hamilton cycle, Hamilton path, n-HC-extendable, strongly n-HP-extendable, weakly n-HP-extendable, Cayley graph, abelian group Objavljeno v RUP: 15.10.2013; Ogledov: 2832; Prenosov: 143 Povezava na celotno besedilo |
3. 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. Ključne besede: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Objavljeno v RUP: 15.10.2013; Ogledov: 3444; Prenosov: 40 Povezava na celotno besedilo |
4. Recent trends and future directions in vertex-transitive graphsKlavdija Kutnar, Dragan Marušič, 2008, izvirni znanstveni članek Opis: A graph is said to be vertex-transitive if its automorphism group acts transitively on the vertex set. Some recent developments and possible future directions regarding two famous open problems, asking about existence of Hamilton paths and existence of semiregular automorphisms in vertex-transitive graphs, are discussed, together with some recent results on arc-transitive graphs and half-arc-transitive graphs, two special classes of vertex-transitive graphs that have received particular attention over the last decade. Ključne besede: vertex-transitive graph, arc-transitive graph, half-arc-transitive graph, Hamilton cycle, Hamilton path, semiregular group, (im)primitive group Objavljeno v RUP: 15.10.2013; Ogledov: 3181; Prenosov: 132 Celotno besedilo (183,49 KB) |
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6. Hamilton paths in vertex-transitive graphs of order 10pKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, izvirni znanstveni članek Opis: It is shown that every connected vertex-transitive graph of order ▫$10p$▫, ▫$p \ne 7$▫ a prime, which is not isomorphic to a quasiprimitive graph arising from the action of PSL▫$(2,k)$▫ on cosets of ▫$\mathbb{Z}_k \times \mathbb{Z}_{(k-1)/10}$▫, contains a Hamilton path. Ključne besede: graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Objavljeno v RUP: 15.10.2013; Ogledov: 3596; Prenosov: 40 Povezava na celotno besedilo |