The Isomorphism Classes of All Generalized Petersen ... - UP FAMNIT
The Isomorphism Classes of All Generalized Petersen Graphs Ted Dobson Department of Mathematics & Statistics Mississippi State University [email protected] http://www2.msstate.edu/∼dobson/
Seminar, University of Primorska, May 9, 2011
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n, k) to be the graph with vertex set Z2 × Zn and edge set {(0, i)(0, i + 1), (0, i)(1, i), (1, i)(1, i + k) : i ∈ Zn }
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n, k) to be the graph with vertex set Z2 × Zn and edge set {(0, i)(0, i + 1), (0, i)(1, i), (1, i)(1, i + k) : i ∈ Zn } The Petersen graph is GP(5, 2):
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n, k) to be the graph with vertex set Z2 × Zn and edge set {(0, i)(0, i + 1), (0, i)(1, i), (1, i)(1, i + k) : i ∈ Zn } The Petersen graph is GP(5, 2): (0, 0) •
•(1, 0) (0, 4)•
•(0, 1)
(1, 4) •
(1, 3) •
• (0, 3)
• (1, 1)
(1, 2)• • (0, 2)
Figure: The Petersen graph is GP(5, 2) Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether “similar” type graphs are Tait colorable.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether “similar” type graphs are Tait colorable.
(0, 9) •
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• (1, 2)
(1, 7) • (1, 6)• • (0, 6)
• (1, 3) • (1, 5)• (1, 4) • (0, 5)
•(0, 2)
•(0, 3)
• (0, 4)
Figure: The generalized Petersen graph GP(10, 4). Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
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Figure: The Desargues configuration and its Levi graph GP(10, 3) Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs:
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
I
GP(8, 3) is the M¨obius-Kantor graph
The generalized Petersen graphs have received a great deal of attention -
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
I
GP(8, 3) is the M¨obius-Kantor graph
The generalized Petersen graphs have received a great deal of attention their automorphism groups are known,
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
I
GP(8, 3) is the M¨obius-Kantor graph
The generalized Petersen graphs have received a great deal of attention their automorphism groups are known, and exactly which contain Hamilton cycles are known for example.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n .
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k),
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k), τ ((1, kj)(1, kj + k)) = (0, ±j)(0, ±j ± 1),
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k), τ ((1, kj)(1, kj + k)) = (0, ±j)(0, ±j ± 1), and τ ((0, j)(1, j)) = (1, kj)(0, kj).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k), τ ((1, kj)(1, kj + k)) = (0, ±j)(0, ±j ± 1), and τ ((0, j)(1, j)) = (1, kj)(0, kj). Thus τ ∈ Aut(GP(n, k)).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi. In 1971, Frucht, Graver, and Watkins determined the automorphism groups of the generalized Petersen graphs:
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi. In 1971, Frucht, Graver, and Watkins determined the automorphism groups of the generalized Petersen graphs:
Theorem Aut(GP(n, k)) = B(n, k) except for the following pairs (n, k), where 2 ≤ 2k < n: (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi. In 1971, Frucht, Graver, and Watkins determined the automorphism groups of the generalized Petersen graphs:
Theorem Aut(GP(n, k)) = B(n, k) except for the following pairs (n, k), where 2 ≤ 2k < n: (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5). They also determined the automorphism groups for the seven exceptional pairs, but we will not discuss them.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Staton and Steimle in 2009 proved the following result:
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Staton and Steimle in 2009 proved the following result:
Theorem For 2 ≤ k ≤ n − 2 with gcd(n, k) = 1, the generalized Petersen graphs, GP(n, k) and GP(n, `) are isomorphic if and only if either k ≡ −` (mod n) or k` ≡ ±1 (mod n).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Staton and Steimle in 2009 proved the following result:
Theorem For 2 ≤ k ≤ n − 2 with gcd(n, k) = 1, the generalized Petersen graphs, GP(n, k) and GP(n, `) are isomorphic if and only if either k ≡ −` (mod n) or k` ≡ ±1 (mod n). Their proof very much makes use of the fact that if gcd(n, k) = 1, then the inside graph is a cycle, and not a disjoint union of cycles.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G .
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G . If one then calculates this normalizer, then the isomorphism problem is solved.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G . If one then calculates this normalizer, then the isomorphism problem is solved. Of course this strategy will only work well if G is transitive and small, or intransitive and large.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G . If one then calculates this normalizer, then the isomorphism problem is solved. Of course this strategy will only work well if G is transitive and small, or intransitive and large. If one is working with Cayley graphs of a group G , then GL is the obvious small transitive subgroup to work with, and this is exactly how the characterization of which groups are CI-groups with respect to graphs was obtained.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For us with the generalized Petersen graphs, the obvious choice of G is hρi, which has two orbits of size n.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For us with the generalized Petersen graphs, the obvious choice of G is hρi, which has two orbits of size n. The normalizer of hρi is
Lemma Let ρ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1). Then NS2n (hρi) = {(i, j) → (i + a, βj + bi ) : a ∈ Z2 , β ∈ Z∗n , bi ∈ Zn }.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For us with the generalized Petersen graphs, the obvious choice of G is hρi, which has two orbits of size n. The normalizer of hρi is
Lemma Let ρ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1). Then NS2n (hρi) = {(i, j) → (i + a, βj + bi ) : a ∈ Z2 , β ∈ Z∗n , bi ∈ Zn }. The only part (sort of) of our strategy remaining is to show that if φ−1 hρiφ ≤ Aut(GP(n, k)), then there exists δ ∈ Aut(GP(n, k)) such that δ −1 φ−1 hρiφδ = hρi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored -
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b) = δ(i, −j)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b) = δ(i, −j) = (i, j)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b) = δ(i, −j) = (i, j) has order 2, not n. Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1,
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1, or contains two maximal cyclic subgroups if gcd(2, n) = 2.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1, or contains two maximal cyclic subgroups if gcd(2, n) = 2. So the case where k = ±1 is finished - GP(n, 1) is only isomorphic to GP(n, n − 1).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1, or contains two maximal cyclic subgroups if gcd(2, n) = 2. So the case where k = ±1 is finished - GP(n, 1) is only isomorphic to GP(n, n − 1). So, in the remaining case, we have an isomorphism φ between GP(n, k) and GP(n, `) that is contained in the normalizer in S2n of hρi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn .
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0. Let O = {(0, j) : j ∈ Zn } (the “outer” vertices of GP(n, k)),
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0. Let O = {(0, j) : j ∈ Zn } (the “outer” vertices of GP(n, k)), and I = {(1, j) : j ∈ Zn } (the “inner” vertices of GP(n, k)).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0. Let O = {(0, j) : j ∈ Zn } (the “outer” vertices of GP(n, k)), and I = {(1, j) : j ∈ Zn } (the “inner” vertices of GP(n, k)). We refer to edges of the form (1, i)(0, i) as spoke edges, and set φ(GP(n, k)) = GP(n, `).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O],
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal). As the automorphism group of cycle is a dihedral group, we conclude that β = ±1.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal). As the automorphism group of cycle is a dihedral group, we conclude that β = ±1. As φ(0, 0) = (0, 0) and φ maps spoke edges to spoke edges, we see that φ(1, 0) = (1, 0) and so b1 = 0 as well.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal). As the automorphism group of cycle is a dihedral group, we conclude that β = ±1. As φ(0, 0) = (0, 0) and φ maps spoke edges to spoke edges, we see that φ(1, 0) = (1, 0) and so b1 = 0 as well. Then φ ∈ hδi, and GP(n, k)[I] = GP(n, `)[I]. Thus ` = ±k.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I].
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I]. As φ((0, 0)(0, 1)) = (1, 0)(1, `) or (1, 0)(1, −`), we have that β = ±`.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I]. As φ((0, 0)(0, 1)) = (1, 0)(1, `) or (1, 0)(1, −`), we have that β = ±`. As φ maps spoke edges to spoke edges and φ(0, 0) = (1, 0), we have that φ(1, 0) = (0, 0) and again b1 = 0.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I]. As φ((0, 0)(0, 1)) = (1, 0)(1, `) or (1, 0)(1, −`), we have that β = ±`. As φ maps spoke edges to spoke edges and φ(0, 0) = (1, 0), we have that φ(1, 0) = (0, 0) and again b1 = 0. Then φ((1, 0)(1, k)) = (0, 0)(0, ±k`) = (0, 0)(0, 1) or (0, 0)(0, −1) and so k` = ±1.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Theorem The generalized Petersen graphs GP(n, k) and GP(n, `) are isomorphic if and only if either k ≡ ±` (mod n) or k` ≡ ±1 (mod n).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Seminar, University of Primorska, May 9, 2011
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n, k) to be the graph with vertex set Z2 × Zn and edge set {(0, i)(0, i + 1), (0, i)(1, i), (1, i)(1, i + k) : i ∈ Zn }
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n, k) to be the graph with vertex set Z2 × Zn and edge set {(0, i)(0, i + 1), (0, i)(1, i), (1, i)(1, i + k) : i ∈ Zn } The Petersen graph is GP(5, 2):
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For integers k and n satisfying 1 ≤ k ≤ n − 1, 2k 6= n, define the generalized Petersen graph GP(n, k) to be the graph with vertex set Z2 × Zn and edge set {(0, i)(0, i + 1), (0, i)(1, i), (1, i)(1, i + k) : i ∈ Zn } The Petersen graph is GP(5, 2): (0, 0) •
•(1, 0) (0, 4)•
•(0, 1)
(1, 4) •
(1, 3) •
• (0, 3)
• (1, 1)
(1, 2)• • (0, 2)
Figure: The Petersen graph is GP(5, 2) Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether “similar” type graphs are Tait colorable.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Generalized Petersen graphs were introduced by Watkins in 1969 who was interested in trivalent graphs without proper three edge-colorings. A proper 3-edge coloring of a trivalent graph is called a Tait coloring. The Petersen graph is not Tait colorable, so it is reasonable to ask whether “similar” type graphs are Tait colorable.
(0, 9) •
(0, 8)•
(0, 7)•
(0, 0) •
(0, 1) • (1, 9) •(1, 0) • •(1, 1)
(1, 8) •
• (1, 2)
(1, 7) • (1, 6)• • (0, 6)
• (1, 3) • (1, 5)• (1, 4) • (0, 5)
•(0, 2)
•(0, 3)
• (0, 4)
Figure: The generalized Petersen graph GP(10, 4). Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
p9
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J`9 Q Q D Q JD p6 `0 Q `6 `8 `3
J D Q
` p4 1 X J p0X Dp Q p5 " " XXX
`4 3 `2 XXX "` "
p1 X X Xp" 5 2
p0 •
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Figure: The Desargues configuration and its Levi graph GP(10, 3) Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs:
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
I
GP(8, 3) is the M¨obius-Kantor graph
The generalized Petersen graphs have received a great deal of attention -
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
I
GP(8, 3) is the M¨obius-Kantor graph
The generalized Petersen graphs have received a great deal of attention their automorphism groups are known,
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Many famous graphs are generalized Petersen graphs: I
GP(5, 2) is the Petersen graph
I
GP(4, 1) is the skeleton of the cube
I
GP(10, 3) is the Desargues graph
I
GP(10, 2) is the skeleton of the dodecahedron
I
GP(8, 3) is the M¨obius-Kantor graph
The generalized Petersen graphs have received a great deal of attention their automorphism groups are known, and exactly which contain Hamilton cycles are known for example.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n .
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k),
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k), τ ((1, kj)(1, kj + k)) = (0, ±j)(0, ±j ± 1),
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k), τ ((1, kj)(1, kj + k)) = (0, ±j)(0, ±j ± 1), and τ ((0, j)(1, j)) = (1, kj)(0, kj).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Automorphisms of generalized Petersen graphs Define ρ, δ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1) and δ(i, j) = (i, −j). It is then easy to see that ρ, δ ∈ Aut(GP(n, k)) for every n and k. One consequence of this is that GP(n, k) = GP(n, −k). Define τ : Z2 × Zn → Z2 × Zn by τ (i, j) = (i + 1, kj). In order for τ to be a bijection, k ∈ Z∗n . In order for τ to be an automorphism of GP(n, k), τ 2 (i, j) = (i, k 2 j) must fix the outside n-cycle and the inside vertices, and map spoke edges to spoke edges. As the automorphism group of an n-cycle is a dihedral group, we conclude that k 2 ≡ ±1 (mod n). If k 2 ≡ ±1, then τ ((0, j)(0, j + 1)) = (1, kj)(1, kj + k), τ ((1, kj)(1, kj + k)) = (0, ±j)(0, ±j ± 1), and τ ((0, j)(1, j)) = (1, kj)(0, kj). Thus τ ∈ Aut(GP(n, k)).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi. In 1971, Frucht, Graver, and Watkins determined the automorphism groups of the generalized Petersen graphs:
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi. In 1971, Frucht, Graver, and Watkins determined the automorphism groups of the generalized Petersen graphs:
Theorem Aut(GP(n, k)) = B(n, k) except for the following pairs (n, k), where 2 ≤ 2k < n: (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If k 2 ≡ 1 (mod n), then set B(n, k) = hρ, δ, τ i, if k 2 ≡ −1 (mod n), set B(n, k) = hρ, τ i, while if k 2 6≡ ±1 (mod n), set B(n, k) = hρ, δi. In 1971, Frucht, Graver, and Watkins determined the automorphism groups of the generalized Petersen graphs:
Theorem Aut(GP(n, k)) = B(n, k) except for the following pairs (n, k), where 2 ≤ 2k < n: (4, 1), (5, 2), (8, 3), (10, 2), (10, 3), (12, 5), (24, 5). They also determined the automorphism groups for the seven exceptional pairs, but we will not discuss them.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Staton and Steimle in 2009 proved the following result:
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Staton and Steimle in 2009 proved the following result:
Theorem For 2 ≤ k ≤ n − 2 with gcd(n, k) = 1, the generalized Petersen graphs, GP(n, k) and GP(n, `) are isomorphic if and only if either k ≡ −` (mod n) or k` ≡ ±1 (mod n).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Staton and Steimle in 2009 proved the following result:
Theorem For 2 ≤ k ≤ n − 2 with gcd(n, k) = 1, the generalized Petersen graphs, GP(n, k) and GP(n, `) are isomorphic if and only if either k ≡ −` (mod n) or k` ≡ ±1 (mod n). Their proof very much makes use of the fact that if gcd(n, k) = 1, then the inside graph is a cycle, and not a disjoint union of cycles.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G .
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G . If one then calculates this normalizer, then the isomorphism problem is solved.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G . If one then calculates this normalizer, then the isomorphism problem is solved. Of course this strategy will only work well if G is transitive and small, or intransitive and large.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
A general strategy to solve an isomorphism problem Suppose that Γ and Γ0 are isomorphic graphs with φ and isomorphism, and G ≤ Aut(Γ) ∩ Aut(Γ0 ). Then φ−1 G φ ≤ Aut(Γ). If there exists δ ∈ Aut(Γ) such that δ −1 φ−1 G φδ = G , then φδ is an isomorphism from Γ to Γ0 that normalizes G . If one then calculates this normalizer, then the isomorphism problem is solved. Of course this strategy will only work well if G is transitive and small, or intransitive and large. If one is working with Cayley graphs of a group G , then GL is the obvious small transitive subgroup to work with, and this is exactly how the characterization of which groups are CI-groups with respect to graphs was obtained.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For us with the generalized Petersen graphs, the obvious choice of G is hρi, which has two orbits of size n.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For us with the generalized Petersen graphs, the obvious choice of G is hρi, which has two orbits of size n. The normalizer of hρi is
Lemma Let ρ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1). Then NS2n (hρi) = {(i, j) → (i + a, βj + bi ) : a ∈ Z2 , β ∈ Z∗n , bi ∈ Zn }.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
For us with the generalized Petersen graphs, the obvious choice of G is hρi, which has two orbits of size n. The normalizer of hρi is
Lemma Let ρ : Z2 × Zn → Z2 × Zn by ρ(i, j) = (i, j + 1). Then NS2n (hρi) = {(i, j) → (i + a, βj + bi ) : a ∈ Z2 , β ∈ Z∗n , bi ∈ Zn }. The only part (sort of) of our strategy remaining is to show that if φ−1 hρiφ ≤ Aut(GP(n, k)), then there exists δ ∈ Aut(GP(n, k)) such that δ −1 φ−1 hρiφδ = hρi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored -
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b) = δ(i, −j)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b) = δ(i, −j) = (i, j)
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
First, the seven exceptionally pairs (n, k) can be ignored - as isomorphic graphs have isomorphic automorphism groups these generalized Petersen graphs GP(n, k) are only isomorphic to GP(n, −k). Second, show that if k 6= ±1, then hρi is the unique maximal cyclic subgroup of B(n, k). In the case where k 2 6= ±1 (and B(n, k) = hρ, δi), any element of B(n, k) can be written as δ a ρb , a ∈ Z2 , b ∈ Zn as hρi/hρ, δi. If a = 0, then δ a ρb ∈ hρi. If a = 1, then (δρb )2 (i, j) = δρb δ(i, j + b) = δρb (i, −j − b) = δ(i, −j) = (i, j) has order 2, not n. Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1,
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1, or contains two maximal cyclic subgroups if gcd(2, n) = 2.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1, or contains two maximal cyclic subgroups if gcd(2, n) = 2. So the case where k = ±1 is finished - GP(n, 1) is only isomorphic to GP(n, n − 1).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
The case where k 2 = ±1 but k 6= ±1, is very similar, with the computations being slightly more complicated. The case where k 2 = ±1 but k ± 1 are Cayley graphs of Z2 × Zn , and either contains a cyclic subgroup of order 2n if gcd(2, n) = 1, or contains two maximal cyclic subgroups if gcd(2, n) = 2. So the case where k = ±1 is finished - GP(n, 1) is only isomorphic to GP(n, n − 1). So, in the remaining case, we have an isomorphism φ between GP(n, k) and GP(n, `) that is contained in the normalizer in S2n of hρi.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn .
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0. Let O = {(0, j) : j ∈ Zn } (the “outer” vertices of GP(n, k)),
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0. Let O = {(0, j) : j ∈ Zn } (the “outer” vertices of GP(n, k)), and I = {(1, j) : j ∈ Zn } (the “inner” vertices of GP(n, k)).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Set φ(i, j) = (i + a, βj + bi ), a ∈ Z2 , β ∈ Z∗n , and bi ∈ Zn . As ρ ∈ B(n, k), we can and do assume without loss of generality that b0 = 0. Let O = {(0, j) : j ∈ Zn } (the “outer” vertices of GP(n, k)), and I = {(1, j) : j ∈ Zn } (the “inner” vertices of GP(n, k)). We refer to edges of the form (1, i)(0, i) as spoke edges, and set φ(GP(n, k)) = GP(n, `).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O],
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal). As the automorphism group of cycle is a dihedral group, we conclude that β = ±1.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal). As the automorphism group of cycle is a dihedral group, we conclude that β = ±1. As φ(0, 0) = (0, 0) and φ maps spoke edges to spoke edges, we see that φ(1, 0) = (1, 0) and so b1 = 0 as well.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 0, then φ maps GP(n, k)[O] to GP(n, `)[O], and these graphs are equal (as the outside cycles of a generalized Petersen graph are always equal). As the automorphism group of cycle is a dihedral group, we conclude that β = ±1. As φ(0, 0) = (0, 0) and φ maps spoke edges to spoke edges, we see that φ(1, 0) = (1, 0) and so b1 = 0 as well. Then φ ∈ hδi, and GP(n, k)[I] = GP(n, `)[I]. Thus ` = ±k.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I].
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I]. As φ((0, 0)(0, 1)) = (1, 0)(1, `) or (1, 0)(1, −`), we have that β = ±`.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I]. As φ((0, 0)(0, 1)) = (1, 0)(1, `) or (1, 0)(1, −`), we have that β = ±`. As φ maps spoke edges to spoke edges and φ(0, 0) = (1, 0), we have that φ(1, 0) = (0, 0) and again b1 = 0.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
If a = 1, then φ maps GP(n, k)[O] to GP(n, `)[I]. As φ((0, 0)(0, 1)) = (1, 0)(1, `) or (1, 0)(1, −`), we have that β = ±`. As φ maps spoke edges to spoke edges and φ(0, 0) = (1, 0), we have that φ(1, 0) = (0, 0) and again b1 = 0. Then φ((1, 0)(1, k)) = (0, 0)(0, ±k`) = (0, 0)(0, 1) or (0, 0)(0, −1) and so k` = ±1.
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
Theorem The generalized Petersen graphs GP(n, k) and GP(n, `) are isomorphic if and only if either k ≡ ±` (mod n) or k` ≡ ±1 (mod n).
Ted Dobson The Isomorphism Classes of All Generalized Petersen Graphs
Mississippi State University
