On orders of automorphisms of vertex-transitive graphs

Potočnik, Primož (author), Toledo, Micael (author), Verret, Gabriel (author)

In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $\Gamma$, $\Gamma \not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\langle g \rangle$ of length equal to the order of $g$. Moreover, we prove that in this case either $\Gamma$ belongs to a well understood family of exceptional graphs or at least $5/12$ of the vertices of $\Gamma$ belong to a regular orbit of $g$. Finally, we give an upper bound on the number of orbits of a cyclic group of automorphisms $C$ of a connected $3$-valent vertex-transitive graph $\Gamma$ in terms of the number of vertices of $\Gamma$ and the length of a longest orbit of $C$.

• Language: English
• Keywords: graphs, automorphism groups, vertex-transitive, regular orbit, cubic, tetravalent
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2024
• Number of pages: Str. 123-153
• Numbering: Vol. 166
• PID: 20.500.12556/RUL-154511
• UDC: 519.17
• ISSN on article: 0095-8956
• DOI: 10.1016/j.jctb.2024.01.001
• COBISS.SI-ID: 182607619
• Publication date in RUL: 19.02.2024

Record is a part of a journal

• Title: Journal of combinatorial theory
• Shortened title: J. comb. theory, Ser. B
• Publisher: Elsevier
• ISSN: 0095-8956
• COBISS.SI-ID: 25721600

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• Link: http://creativecommons.org/licenses/by-nc-nd/4.0/
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