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The Hamiltonicity of Subgroup Graphs
Immanuel McLaughlin
Andrew Owens
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• A graph is a set of vertices, V, and a set of edges, E, (denoted by {v1,v2}) where v1,v2 V and {v1,v2} E if there is a line between v1 and v2.
• A subgroup graph of a group G is a graph where the set of vertices is all subgroups of G and the set of edges connects a subgroup to a supergroup if and only if there are no intermediary subgroups.
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Examples of Subgroup Graphs
Zp3
Zp2
Zp
Q8
<i> <j> <k>
<-1>
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• A graph is bipartite if the set of vertices V can be broken into two subset V1 and V2 where there are no edges connecting any two vertices of the same subset.
Definitions
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Examples of Bipartite Graphs
Zp3
Zp2
Zp
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Graph Cartesian Products
• Let G and H be graphs, then the vertex set of G x H is V(G) x V(H).
• An edge, {(g,h),(g`,h`)}, is in the edge set of G x H if g = g` and h is adjacent to h` or h = h` and g is adjacent to g`.
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Examples of Graph Product
x =
x =
1
2
1
2
3
(1,1)
(2,1)
(1,3)
(2,3)
(2,2)
(1,2)
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Results on Graph Products
• The graph product of two bipartite graphs is bipartite.
• The difference in the size of the partitions of a graph product is the product of the difference in the size of the partitions of each graph in the product.
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• Unbalanced bipartite graphs are never Hamiltonian. The reverse is not true in general.
< e >
Zp2
Zp
Zp2
q
Zpq
Zq
Zp2
q2
Zpq2
Zq2
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• For two relatively prime groups, G1 and G2, the subgroup graph of G1 X G2 is isomorphic to the graph cartesian product of the subgroup graphs of G1 and G2.
• The fundamental theorem of finite abelian groups says that every group can be represented as the cross product of cyclic p-groups.
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Finite Abelian Groups
• Finite abelian p-groups are balanced if and only if where n is odd.nG p
Z3 x Z3
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Finite Abelian Groups
• A finite abelian group is balanced if and only if when decomposed into p-groups
x … x , is odd for some .11p
G nnp
G i i
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Cyclic Groups
• Cyclic p-groups are nonhamiltonian.
• Cyclic groups, , with more than one prime factor are hamiltonian if and only if there is at least one that is odd.
11 ... n
npp Z
i
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Cyclic Groups
Zp3
< e >
Zp2
Zp
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Cyclic Groups
Zp3
< e >
Zp2
Zp
Zp3
q2
Zp3
q
Zp3
< e >
Zp2
Zp
Zp2
q
Zpq
Zq
Zp2
q2
Zpq2
Zq2
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Cyclic Groups
Zp3
< e >
Zp2
Zp
Zq
Zq2
Zp3
q2
r2
Zr2Zr
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x is nonhamiltonian. p
ZpZ
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(1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1) (1,0,1)
(0,0,1) (1,0,0) (1,1,0) (1,1,1) (0,1,1) (1,0,1) (0,1,0)
Z2 x Z2 x Z2
2 2 2 Z Z Z
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p p p Z Z Z0 0 1 1 0
1 0 1 0 1
0 1 0 0 0
1
0 0 1 1 1
1 0 1 0 1
0 1 0 0 1
5
0 0 1 1 0
1 0 1 0 1
0 1 0 0 1
4
0 0 1 1 1
1 0 1 0 0
0 1 0 0 1
6
0 0 1 1 1
1 0 1 0 0
0 1 0 0 0
7
0 0 1 1 1
1 0 1 0 1
0 1 0 0 0
3
0 0 1 1 0
1 0 1 0 0
0 1 0 0 1
2
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(1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1) (1,0,1)
(0,0,1) (1,0,0) (1,1,0) (1,1,1) (0,1,1) (1,0,1) (0,1,0)
Z2 x Z2 x Z2
2 2 2 Z Z Z
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(1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1) (1,0,1)
(0,0,1) (1,0,0) (1,1,0) (1,1,1) (0,1,1) (1,0,1) (0,1,0)
Z2 x Z2 x Z2
2 2 2 Z Z Z
(1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1) (1,0,1)
(0,0,1) (1,0,0) (1,1,0) (1,1,1) (0,1,1) (1,0,1) (0,1,0)
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(1,0,0) (0,1,0) (0,0,1) (1,1,0) (0,1,1) (1,1,1) (1,0,1)
(0,0,1) (1,0,0) (1,1,0) (1,1,1) (0,1,1) (1,0,1) (0,1,0)
(1,0,0)
(0,0,1)
Z2 x Z2 x Z2
2 2 2 Z Z Z
(1,0,0)
(0,0,1)
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Dihedral Groups
• Dihedral groups are bipartite and the difference in the size of the partitions of
is , where
1 2
1 1
1 21
( 1)( )
1
i i
n
ki
ni i
pp p p
p
1 21 22 n
np p pD
1 if x is square( )
0 otherwisex
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D12D12
Z6
Z3Z2
D6 D6 D4 D4 D4
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A4
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S4