On certain edge-transitive bicirculants
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2018
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| _version_ | 1866916509385752576 |
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| author | Jajcay, Robert Miklavič, Štefko Šparl, Primož Vasiljević, Gorazd |
| author_facet | Jajcay, Robert Miklavič, Štefko Šparl, Primož Vasiljević, Gorazd |
| contents | A graph $Γ$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabačjn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabačjn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_1801_01106 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | On certain edge-transitive bicirculants Jajcay, Robert Miklavič, Štefko Šparl, Primož Vasiljević, Gorazd Combinatorics 05C25 A graph $Γ$ of even order is a bicirculant if it admits an automorphism with two orbits of equal length. Symmetry properties of bicirculants, for which at least one of the induced subgraphs on the two orbits of the corresponding semiregular automorphism is a cycle, have been studied, at least for the few smallest possible valences. For valences $3$, $4$ and $5$, where the corresponding bicirculants are called generalized Petersen graphs, Rose window graphs and Tabačjn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabačjn graphs, infinite families of edge-transitive Rose window graphs exist. The main theme of this paper is the question of the existence of such bicirculants for higher valences. It is proved that infinite families of edge-transitive examples of valence $6$ exist and among them infinitely many arc-transitive as well as infinitely many half-arc-transitive members are identified. Moreover, the classification of the ones of valence $6$ and girth $3$ is given. As a corollary, an infinite family of half-arc-transitive graphs of valence $6$ with universal reachability relation, which were thus far not known to exist, is obtained. |
| title | On certain edge-transitive bicirculants |
| topic | Combinatorics 05C25 |
| url | https://arxiv.org/abs/1801.01106 |