Hypercube problems
Lecture 2
October 17, 2012
Lecturer: Petr Gregor
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Scribe by: Václav Vlček
Updated: November 6, 2012
Automorphism group
In this lecture we focus on the group of automorphisms of the hypercube. We start with
some algebraic notions that will be useful. Let G = (V, E) be a graph. The group of all
permutations of V is called the symmetric group on V and is denoted by Sym(V).
Definition 1 An automorphism of G is a bijection f : V → V such that uv ∈ E if and
only if f (u)f (v) ∈ E.
Notice that all automorphisms of G form a group with the composition as the binary
operation, the inverse automorphism as the inverse and the identity as the neutral element.
This group is denoted by Aut(G).
Clearly, Aut(G) ≤ Sym(V), i.e. automorphisms form a subgroup of the symmetric
group. There is equality if and only if G is a graph without edges or G is a complete
graph. Furthermore, it follows directly from the definitions that Aut(G) = Aut(G) where
G = (V, E) is the complement of G.
Definition 2 An action of a group Γ on a set V is a homomorphism ϕ : Γ → Sym(V ). If
the action ϕ is fixed, we write g(x) instead of ϕ(g)(x) for g ∈ Γ, x ∈ V and we say that Γ
acts on V .
Aut(G) acts naturally on V by taking identity as its action ϕ : Aut(G) → Sym(V ). In
the following, this action of Aut(G) on V is fixed.
Definition 3 A group Γ acts transitively on a set V if for every x, y ∈ V there is g ∈ Γ
such that g(x) = y. A graph G is vertex-transitive if Aut(G) acts transitively on V .
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Cayley graphs
In the last lecture we saw that the hypercube can be defined as the Cayley graph Qn =
Cay(Zn2 , {e1 , . . . , en }). We will show that every Cayley graph is vertex-transitive (and consequently every hypercube is transitive as well).
We recall the definition of (undirected) Cayley graphs. Let Γ be a group and S ⊆ Γ be
a set such that e 6∈ S and s ∈ S ⇒ s−1 ∈ S. Then the Cayley graph on Γ generated by S is
Cay(Γ, S) = (Γ, {xy | yx−1 ∈ S}).
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Theorem 4 Every Cayley graph G = Cay(Γ, S) is vertex-transitive.
Proof For each g ∈ Γ (i.e. a vertex of G) we define a map πg : Γ → Γ by πg : x 7→ xg.
Note that πg is bijective since πg (yg −1 ) = y for every y ∈ Γ.
Let xy ∈ E(G); that is, according to the definition, yx−1 ∈ S. For every g ∈ Γ,
yx−1 = ygg −1 x−1 = (yg)(xg)−1
so (yg)(xg)−1 ∈ S, i.e. (xg)(yg) ∈ E(G). That means that πg is an automorphism of G.
Since {πg | g ∈ Γ} is closed under composition and contains identity, it forms a subgroup
of Aut(G). Moreover, it acts transitively on Γ as for every x, y ∈ Γ we have πx−1 y (x) = y.
Since Aut(G) has a transitive subgroup, it is transitive as well.
Corollary 5 Qn is vertex-transitive for every n ≥ 1.
There are many open questions regarding vertex-transitive graphs. One of the famous
is the following conjecture.
Conjecture 6 (Lovász [2]) Every connected vertex-transitive graph has a Hamilton path.
The conjecture is open even for many subclasses of connected vertex-transitive graphs,
including connected Cayley graphs. Moreover, there are only five known examples of vertextransitive graphs that have a Hamilton path but do not have a Hamilton cycle (K2 , the
Peterson graph, the Coxeter graph, and two graphs obtained from the Petersen and Coxeter
graphs by replacing each vertex with a triangle [1]).
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Orbit-stabilizer theorem
Definition 7 (stabilizer) Let Γ be a group acting on a set V and let x ∈ V . A stabilizer
of x is the set Γx = {g ∈ Γ | g(x) = x}.
Thus, a stabilizer of x contains precisely the elements whose action fixes x. It is obvious
that Γx is a subgroup of Γ acting on V .
Definition 8 (orbit) Let Γ be a group acting on a set V and let x ∈ V . An orbit of x is
the set Γx = {g(x) | g ∈ Γ}.
In other words, an orbit is a set of all points of V that are images of x under action
of some elements from Γ. We can notice that y ∈ Γx if and only if Γx = Γy (because of
the existence of inverses). Thus orbits give a partition of V . In addition, it follows directly
from the definition that Γ is transitive if and only if there is only one orbit in Γ.
Lemma 9 Let S ⊆ V be an orbit of Γ acting on V and let x, y ∈ S. Then {g ∈ Γ | g(x) = y}
is a right coset of Γx . Conversely, all elements in a right coset of Γx map x to the same
point in S.
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Proof Since x, y are in the same orbit, there is h ∈ Γ such that h(x) = y. If g(x) = y
for g ∈ Γ, then gh−1 (x) = x. Thus gh−1 ∈ Γx , i.e. g ∈ Γx h. That is, g is in the right coset
Γx h.
For the other direction, let us assume that g is arbitrary from the right coset Γx h for a
fixed h, i.e. gh−1 ∈ Γx . Then g(x) = gh−1 h(x) = h(x), where the last equality is from the
fact that gh−1 ∈ Γx (i.e. it stabilizes x).
Theorem 10 (orbit-stabilizer) Let Γ be a group acting on a set V and let x ∈ V . Then
|Γx | · |Γx| = |Γ|
Proof By Lemma 9, there is a one-to-one correspondence between points of the orbit Γx
and the right cosets of the stabilizer Γx . Therefore the group Γ can be partitioned into |Γx|
cosets, each of size |Γx | which gives use the statement.
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Automorphisms of hypercubes
In the rest, we study the automorphism group of hypercubes. We will define two types of
automorphisms and we prove that there are no other automorphism then their compositions.
Definition 11 For n ≥ 1, a ∈ Zn2 , and π ∈ Sn (the symmetric group on [n])
• a translation by a is the map ta : u 7→ u ⊕ a,
• a rotation by π is the map rπ : u 7→ uπ where uπ = (uπ(1) , . . . , uπ(n) ) for u =
(u1 , . . . , un ).
We can view a translation as an operation that flips some fixed bits, or geometrically, it
flips the hypercube. In fact, we have already encountered transitions ta as πa in the proof
n
of Theorem 4. Clearly, ta · tb = ta⊕b , t−1
a = ta and t0 = id. Therefore, Tn = {ta | a ∈ Z2 } is
n
a subgroup of Aut(Qn ) and Tn ' Z2 .
As it concerns rotations, rπ · rρ = rπ·ρ , rπ−1 = rπ−1 and rid = id. Therefore Rn is a
subgroup of Aut(Qn ) as well and Rn ' Sn .
Observation 12 Γx ' Rn for every x ∈ V (Qn ) and Γ = Aut(Qn ).
Proof We show that every g ∈ Γx is uniquely determined by g N (x); that is a permutation of neighbors of x extends uniquely to an automorphism of Qn that fixes x. We
proceed by levels from x, starting by level 2. Let y be in level l ≥ 2. It has at least two
neighbors u1 , u2 in level l − 1. By (0, 2)-property, u1 and u2 have exactly one another
common neighbor z = u1 ⊕ u2 ⊕ y in level l − 2. If g is an automorphism of Qn , it maps
g(y) = g(u1 ) ⊕ g(u2 ) ⊕ g(z). Thus, if g is uniquely extended to z, u1 and u2 , its extension
to y is also unique.
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Corollary 13 |Aut(Qn )| = n!2n .
Proof Since |Γx | = n! by Observation 12 and |Γx| = |V (Qn )| = 2n by Corollary 5, the
statement follows from the orbit-stabilizer theorem.
Corollary 14 For every g ∈ Aut(Qn ) there is a unique π ∈ Sn and a unique a ∈ Zn2 such
that g = gπ,a = rπ · ta .
Proof It is clear that Rn ∩ Tn = {id}. Furthermore, every automorphism g ∈ Rn ·
tn ⊆ Aut(Qn ) is defined uniquely as g = rπ · ta . By Corollary 13, there are no other
automorphisms.
Notes
It remains to determine the structure of Aut(Qn ). This will be done in the next lecture.
The present lecture is (in most) covered by any textbook on algebraic graph theory. The
treatment here is based on [1] which is recommended for further reading.
References
[1] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2004.
[2] L. Lovász, Problem 11 in Combinatorial Structures and their Applications, Gordon
and Breach Science Publishers, New York, 1970.
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