On orders of automorphisms of vertex-transitive graphs

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

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 $C$.

• Jezik: Angleški jezik
• Ključne besede: graphs, automorphism groups, vertex-transitive, regular orbit, cubic, tetravalent
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Datum objave: 01.05.2024
• Leto izida: 2024
• Št. strani: Str. 123-153
• Številčenje: Vol. 166
• PID: 20.500.12556/RUL-154511
• UDK: 519.17
• ISSN pri članku: 0095-8956
• DOI: 10.1016/j.jctb.2024.01.001
• COBISS.SI-ID: 182607619
• Datum objave v RUL: 19.02.2024

Gradivo je del revije

• Naslov: Journal of combinatorial theory
• Skrajšan naslov: J. comb. theory, Ser. B
• Založnik: Academic Press
• ISSN: 0095-8956
• COBISS.SI-ID: 25721600

Licence

• Licenca: CC BY-NC-ND 4.0, Creative Commons Priznanje avtorstva-Nekomercialno-Brez predelav 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by-nc-nd/4.0/deed.sl
• Opis: Najbolj omejujoča licenca Creative Commons. Uporabniki lahko prenesejo in delijo delo v nekomercialne namene in ga ne smejo uporabiti za nobene druge namene.