Abstract
A graph $\Gamma$ 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 Tabač jn graphs, respectively, all edge-transitive members have been classified. While there are only 7 edge-transitive generalized Petersen graphs and only 3 edge-transitive Tabač jn 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.
Highlights
IntroductionEven though almost all graphs have no nontrivial automorphisms (see for instance [7, Corollary 2.3.3]) investigation of highly symmetric graphs has been a very active topic of research in algebraic graph theory for decades
Even though almost all graphs have no nontrivial automorphisms investigation of highly symmetric graphs has been a very active topic of research in algebraic graph theory for decades
In 1981 Marusic conjectured [18] that every vertex-transitive graph admits a nontrivial semiregular automorphism
Summary
Even though almost all graphs have no nontrivial automorphisms (see for instance [7, Corollary 2.3.3]) investigation of highly symmetric graphs has been a very active topic of research in algebraic graph theory for decades. It should be pointed out that the members of the infinite family of graphs that has quite recently been obtained by Zhou and Zhang [26] when they classified half-arc-regular bicirculants of valence 6 turn out to be Nest graphs. For which integers d > 6 does there exist an edge-transitive bicirculant of valence d, such that at least one of the subgraphs induced on the two orbits of the corresponding semiregular automorphism is a cycle? An exhaustive computer search shows that there exists no edge-transitive bicirculant of valence d where 7 d 10 and order at most 100, such that at least one of the subgraphs induced on the two orbits of the corresponding semiregular automorphism is a cycle. In Theorem 23 we prove that 6 is attained by exhibiting an infinite family of half-arc-transitive graphs of valence 6 with universal reachability relation
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