On arc-transitive graphs with large arc-stabilisers - SymOmega

83 Pages·2010·0.22 MB·English

by  Gabriel Verret

On arc-transitive graphs with large arc-stabilisers
Gabriel Verret
Perth, March 16th, 2010
(cid:73) Let Γ be such a graph and let G ≤ Aut(Γ) be a group of
automorphisms of Γ.
(cid:73) We say that Γ is G-arc-transitive if G acts transitively on arcs
of Γ.
Definitions
(cid:73) All graphs considered will be finite and connected.
(cid:73) We say that Γ is G-arc-transitive if G acts transitively on arcs
of Γ.
Definitions
(cid:73) All graphs considered will be finite and connected.
(cid:73) Let Γ be such a graph and let G ≤ Aut(Γ) be a group of
automorphisms of Γ.
Definitions
(cid:73) All graphs considered will be finite and connected.
(cid:73) Let Γ be such a graph and let G ≤ Aut(Γ) be a group of
automorphisms of Γ.
(cid:73) We say that Γ is G-arc-transitive if G acts transitively on arcs
of Γ.
(cid:73) Practical interest : Foster Census of 3-valent arc-transitive
graphs of small order (Conder and Dobcs´anyi). Their
approach is dependent on Tutte’s Theorem.
(cid:73) We would like to generalize this result.
The 3-valent case
Theorem (Tutte 1946)
If Γ is a 3-valent G-arc-transitive graph, then G acts regularly on
s-arcs for some s ≤ 5. In particular, |G | = 2s−1 ≤ 24.
uv
(cid:73) We would like to generalize this result.
The 3-valent case
Theorem (Tutte 1946)
If Γ is a 3-valent G-arc-transitive graph, then G acts regularly on
s-arcs for some s ≤ 5. In particular, |G | = 2s−1 ≤ 24.
uv
(cid:73) Practical interest : Foster Census of 3-valent arc-transitive
graphs of small order (Conder and Dobcs´anyi). Their
approach is dependent on Tutte’s Theorem.
The 3-valent case
Theorem (Tutte 1946)
If Γ is a 3-valent G-arc-transitive graph, then G acts regularly on
s-arcs for some s ≤ 5. In particular, |G | = 2s−1 ≤ 24.
uv
(cid:73) Practical interest : Foster Census of 3-valent arc-transitive
graphs of small order (Conder and Dobcs´anyi). Their
approach is dependent on Tutte’s Theorem.
(cid:73) We would like to generalize this result.
n
(cid:73) If G = Aut(W(m,2)), then |Gv| = 22, where
n = |V(W(m,2))|.
(cid:73) Not that the bound is exponential with respect to the number
of vertices.
Wreath graphs
(cid:73) Consider the wreath graph W(m,2) ∼= C [K¯ ].
m 2
(cid:73) Not that the bound is exponential with respect to the number
of vertices.
Wreath graphs
(cid:73) Consider the wreath graph W(m,2) ∼= C [K¯ ].
m 2
n
(cid:73) If G = Aut(W(m,2)), then |Gv| = 22, where
n = |V(W(m,2))|.
Wreath graphs
(cid:73) Consider the wreath graph W(m,2) ∼= C [K¯ ].
m 2
n
(cid:73) If G = Aut(W(m,2)), then |Gv| = 22, where
n = |V(W(m,2))|.
(cid:73) Not that the bound is exponential with respect to the number
of vertices.