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 1 - 3 / 31 1.Hamilton cycles in (2, odd, 3)-Cayley graphsHenry Glover, Klavdija Kutnar, Aleksander Malnič, Dragan Marušič, 2012, original scientific articleAbstract: 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 besedahSummary of found: ...Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism group...Keywords: Cayley graph, Hamilton cycle, arc-transitive graph, 1-regular action, automorphism groupPublished: 15.10.2013; Views: 1254; Downloads: 61 Full text (0,00 KB) 2.Algebraični aspekti teorije grafovAdemir Hujdurović, 2013, doctoral dissertationFound in: ključnih besedahSummary of found: ...quasi m-Cayley graph, generalized Cayley graph, I-regular action, regular cover of a graph, automorphism group,... ...circulant, bicirculant, semiregular automorphism, vertex-transitive graph, half-arc-transitive graph, snark, Cayley graph,...Keywords: circulant, bicirculant, semiregular automorphism, vertex-transitive graph, half-arc-transitive graph, snark, Cayley graph, quasi m-Cayley graph, generalized Cayley graph, I-regular action, regular cover of a graph, automorphism groupPublished: 10.07.2015; Views: 1569; Downloads: 4 Full text (0,00 KB) 3.On even-closed regular embeddings of graphsIstván Kovács, Klavdija Kutnar, Dragan Marušič, Daniel Pellicer, 2018, original scientific articleFound in: ključnih besedahSummary of found: ...odd action automorphism, even action automorphism, automorphism group, vertex-transitive...Keywords: odd action automorphism, even action automorphism, automorphism group, vertex-transitive graph, orientable-regular mapPublished: 18.07.2018; Views: 937; Downloads: 84 Full text (0,00 KB)
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