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1.
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 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: 1254; Downloads: 61
URL Full text (0,00 KB)

2.
Algebraični aspekti teorije grafov
Ademir Hujdurović, 2013, doctoral dissertation

Found in: ključnih besedah
Summary 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 group
Published: 10.07.2015; Views: 1569; Downloads: 4
URL Full text (0,00 KB)

3.
On even-closed regular embeddings of graphs
István Kovács, Klavdija Kutnar, Dragan Marušič, Daniel Pellicer, 2018, original scientific article

Found in: ključnih besedah
Summary 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 map
Published: 18.07.2018; Views: 937; Downloads: 84
URL Full text (0,00 KB)

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