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: ...It is shown that every connected vertex-transitive graph of order ▫$10p$▫, ▫$p \ne 7$▫ a... ...graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group... Keywords: graph, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Published: 15.10.2013; Views: 1778; Downloads: 12 Full text (0,00 KB) |
2. On prime-valent symmetric bicirculants and Cayley snarksAdemir Hujdurović, Klavdija Kutnar, Dragan Marušič, 2013, published scientific conference contribution Found in: ključnih besedah Summary of found: ... graph, Cayley graph, arc-transitive, snark, semiregular automorphism, bicirculant,... Keywords: graph, Cayley graph, arc-transitive, snark, semiregular automorphism, bicirculant Published: 15.10.2013; Views: 1632; Downloads: 81 Full text (0,00 KB) |
3. 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: ...It is shown that every connected vertex-transitive graph of order ▫$4p$▫, where ▫$p$▫ is a... ...graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group... Keywords: graph theory, vertex-transitive graphs, Hamilton cycle, automorphism group Published: 15.10.2013; Views: 1655; Downloads: 18 Full text (0,00 KB) |
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5. On quartic half-arc-transitive metacirculantsDragan 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: ključnih besedah Summary of found: ...Following Alspach and Parsons, a metacirculant graph is a graph admitting a transitive group... ...theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group... Keywords: mathematics, graph theory, metacirculant graph, half-arc-transitive graph, tightly attached, automorphism group Published: 15.10.2013; Views: 1738; Downloads: 68 Full text (0,00 KB) |
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7. Asymptotic automorphism groups of Cayley digraphs and graphs of abelian groups of prime-power orderEdward Dobson, 2010, original scientific article Abstract: 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, ▫$G_L \triangleleft {\rm Aut}(\Gamma))$▫. Found in: ključnih besedah Summary of found: ...We show that almost every Cayley graph ▫$\Gamma$▫ of an abelian group ▫$G$▫ of... Keywords: mathematics, graph theory, Cayley graph, abelian group, automorphism group, asymptotic, ▫$p$▫-group Published: 15.10.2013; Views: 2579; Downloads: 64 Full text (0,00 KB) |
8. 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 Summary of found: ...It is shown that every connected vertex-transitive graph of order ▫$6p$▫, where ▫$p$▫ is a... ...graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group... Keywords: graph theory, vertex-transitive, Hamilton cycle, Hamilton path, automorphism group Published: 15.10.2013; Views: 1717; Downloads: 14 Full text (0,00 KB) |
9. On cubic non-Cayley vertex-transitive graphsKlavdija Kutnar, Dragan Marušič, Cui Zhang, 2012, original scientific article Found in: ključnih besedah Summary of found: ...vertex-transitive graph, non-Cayley graph, automorphism group, ... Keywords: vertex-transitive graph, non-Cayley graph, automorphism group Published: 15.10.2013; Views: 1311; Downloads: 67 Full text (0,00 KB) |
10. 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: ...Lovász asked if every finite, connected vertex-transitive graph has a Hamilton path. In spite of... ...graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group... Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group Published: 15.10.2013; Views: 1279; Downloads: 63 Full text (0,00 KB) |