1. Hamilton paths in vertex-transitive graphs of order 10pKlavdija 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: 1694; Downloads: 11 Full text (0,00 KB) |
2. Hamiltonicity of vertex-transitive graphs of order 4pKlavdija 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: ...▫$4p$▫, where ▫$p$▫ is a prime, is hamiltonian with the exception of the Coxeter graph... ...graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group... Keywords: graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group Published: 15.10.2013; Views: 1580; Downloads: 17 Full text (0,00 KB) |
3. Hamiltonicity of cubic Cayley graphsDragan 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: 1369; Downloads: 25 Full text (0,00 KB) |
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5. Hamilton paths and cycles in vertex-transitive graphs of order 6pKlavdija 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 Keywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Published: 15.10.2013; Views: 1628; Downloads: 14 Full text (0,00 KB) |
6. On Hamiltonicity of circulant digraphs of outdegree threeŠtefko 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: 1273; Downloads: 46 Full text (0,00 KB) |
7. Hamilton cycles in (2, odd, 3)-Cayley graphsHenry 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: 1217; Downloads: 56 Full text (0,00 KB) |
8. Hamilton cycle and Hamilton path extendability of Cayley graphs on abelian groupsŠtefko 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 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: 1224; Downloads: 70 Full text (0,00 KB) |
9. Special issue of discrete mathematics on Hamiltonicity problem for vertex-transitive (Cayley) graphsDragan Marušič, 2009, preface, afterword Found in: ključnih besedah Summary of found: ... Hamilton cycle, Hamilton path, Vetrex-transitive graph, Cayley graph,... Keywords: Hamilton cycle, Hamilton path, Vetrex-transitive graph, Cayley graph Published: 15.10.2013; Views: 1308; Downloads: 13 Full text (0,00 KB) |
10. On cyclic edge-connectivity of fullerenesDragan Marušič, Klavdija Kutnar, 2008, original scientific article Abstract: A graph is said to be cyclically ▫$k$▫-edge-connected, if at least ▫$k$▫ edges must be removed to disconnect it into two components, each containing a cycle. Such a set of ▫$k$▫ edges is called a cyclic-k-edge cutset and it is called a trivial cyclic-k-edge cutset if at least one of the resulting two components induces a single ▫$k$▫-cycle. It is known that fullerenes, that is, 3-connected cubic planar graphs all of whose faces are pentagons and hexagons, are cyclically 5-edge-connected. In this article it is shown that a fullerene ▫$F$▫ containing a nontrivial cyclic-5-edge cutset admits two antipodal pentacaps, that is, two antipodal pentagonal faces whose neighboring faces are also pentagonal. Moreover, it is shown that ▫$F$▫ has a Hamilton cycle, and as a consequence at least ▫$15 \cdot 2^{n/20-1/2}$▫ perfect matchings, where ▫$n$▫ is the order of ▫$F$▫. Found in: ključnih besedah Summary of found: ...it is shown that ▫$F$▫ has a Hamilton cycle, and as a consequence at least... ...it into two components, each containing a cycle. Such a set of ▫$k$▫ edges is... Keywords: graph, fullerene graph, cyclic edge-connectivity, hamilton cycle, perfect matching Published: 03.04.2017; Views: 668; Downloads: 77 Full text (0,00 KB) |