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1.
Hamilton paths in vertex-transitive graphs of order 10p
Klavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific article

Abstract: 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.
Found in: ključnih besedah
Summary of found: ...graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group...
Keywords: graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group
Published: 15.10.2013; Views: 1808; Downloads: 12
URL Full text (0,00 KB)

2.
Hamiltonicity of vertex-transitive graphs of order 4p
Klavdija Kutnar, Dragan Marušič, 2008, original scientific article

Abstract: 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.
Found in: ključnih besedah
Summary of found: ...graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group...
Keywords: graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group
Published: 15.10.2013; Views: 1677; Downloads: 18
URL Full text (0,00 KB)

3.
A complete classification of cubic symmetric graphs of girth 6
Klavdija Kutnar, Dragan Marušič, 2009, original scientific article

Abstract: A complete classification of cubic symmetric graphs of girth 6 is given. It is shown that with the exception of the Heawood graph, the Moebius-Kantor graph, the Pappus graph, and the Desargues graph, a cubic symmetric graph ▫$X$▫ of girth 6 is a normal Cayley graph of a generalized dihedral group; in particular, (i) ▫$X$▫ is 2-regular if and only if it is isomorphic to a so-called ▫$I_k^n$▫-path, a graph of order either ▫$n^2/2$▫ or ▫$n^2/6$▫, which is characterized by the fact that its quotient relative to a certain semiregular automorphism is a path. (ii) ▫$X$▫ is 1-regular if and only if there exists an integer ▫$r$▫ with prime decomposition ▫$r=3^s p_1^{e_1} \dots p_t^{e_t} > 3$▫, where ▫$s \in \{0,1\}$▫, ▫$t \ge 1$▫, and ▫$p_i \equiv 1 \pmod{3}$▫, such that ▫$X$▫ is isomorphic either to a Cayley graph of a dihedral group ▫$D_{2r}$▫ of order ▫$2r$▫ or ▫$X$▫ is isomorphic to a certain ▫$\ZZ_r$▫-cover of one of the following graphs: the cube ▫$Q_3$▫, the Pappus graph or an ▫$I_k^n(t)$▫-path of order ▫$n^2/2$▫.
Found in: ključnih besedah
Summary of found: ...graphs, symmetric graphs, ▫$s$▫-regular graphs, girth, consistent cycle...
Keywords: graph theory, cubic graphs, symmetric graphs, ▫$s$▫-regular graphs, girth, consistent cycle
Published: 15.10.2013; Views: 1884; Downloads: 57
URL Full text (0,00 KB)

4.
Consistent Cycles in 1/2-Arc-Transitive Graphs
Marko Boben, Štefko Miklavič, Primož Potočnik, 2009, original scientific article

Found in: ključnih besedah
Summary of found: ...mathematics, graph theory, 1/2-arc-transitivity, consistent cycle, ...
Keywords: mathematics, graph theory, 1/2-arc-transitivity, consistent cycle
Published: 15.10.2013; Views: 2244; Downloads: 7
URL Full text (0,00 KB)
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5.
Hamiltonicity of cubic Cayley graphs
Dragan Marušič, Henry Glover, Klavdija Kutnar, Aleksander Malnič, 2012, published scientific conference contribution abstract (invited lecture)

Found in: ključnih besedah
Summary of found: ...Cayley graph, Hamilton path, Hamilton cycle, arc-transitive graph, Cayley map, ...
Keywords: Cayley graph, Hamilton path, Hamilton cycle, arc-transitive graph, Cayley map
Published: 15.10.2013; Views: 1457; Downloads: 35
URL Full text (0,00 KB)

6.
Construction of Hamilton cycles in (2,s,3)-Cayley graphs
Klavdija Kutnar, 2010, published scientific conference contribution abstract

Found in: ključnih besedah
Summary of found: ...Hamilton cycle, Cayley graph, ...
Keywords: Hamilton cycle, Cayley graph
Published: 15.10.2013; Views: 1381; Downloads: 8
URL Full text (0,00 KB)

7.
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: ključnih besedah
Summary of found: ...is not genuinely imprimitive contains a Hamilton cycle....
Keywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group
Published: 15.10.2013; Views: 1746; Downloads: 14
URL Full text (0,00 KB)

8.
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: ključnih besedah
Summary of found: ...graph theory, circulant digraph, Hamilton cycle...
Keywords: graph theory, circulant digraph, Hamilton cycle
Published: 15.10.2013; Views: 1371; Downloads: 58
URL Full text (0,00 KB)

9.
Hamilton cycles in (2, odd, 3)-Cayley graphs
Henry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific article

Abstract: In 1969, Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of its easy formulation, no major breakthrough has been achieved thus far, and the problem is now commonly accepted to be very hard. The same holds for the special subclass of Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and Marušič proved that a cubic Cayley graph on a finite ▫$(2, s, 3)$▫-generated group ▫$G = \langle a, x| a^2 = x^s = (ax)^3 = 1, \dots \rangle$▫ has a Hamilton path when ▫$|G|$▫ is congruent to 0 modulo 4, and has a Hamilton cycle when ▫$|G|$▫ is congruent to 2 modulo 4. The Hamilton cycle was constructed, combining the theory of Cayley maps with classical results on cyclic stability in cubic graphs, as the contractible boundary of a tree of faces in the corresponding Cayley map. With a generalization of these methods, Glover, Kutnar and Marušič in 2009 resolved the case when, apart from ▫$|G|$▫, also ▫$s$▫ is congruent to 0 modulo 4. In this article, with a further extension of the above "tree of faces" approach, a Hamilton cycle is shown to exist whenever ▫$|G|$▫ is congruent to 0 modulo 4 and s is odd. This leaves ▫$|G|$▫ congruent to 0 modulo 4 with s congruent to 2 modulo 4 as the only remaining open case. In this last case, however, the "tree of faces" approach cannot be applied, and so entirely different techniques will have to be introduced if one is to complete the proof of the existence of Hamilton cycles in cubic Cayley graphs arising from finite ▫$(2, s, 3)$▫-generated groups.
Found in: ključnih besedah
Summary of found: ...Cayley graphs where the existence of Hamilton cycles has been conjectured. In 2007, Glover and...
Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group
Published: 15.10.2013; Views: 1293; Downloads: 67
URL Full text (0,00 KB)

10.
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: ključnih besedah
Summary of found: Zadetek v naslovu
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: 1318; Downloads: 82
URL Full text (0,00 KB)

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