Toric Graph Ideals
Bryan Brown, Laura Lyman, Amy Nesky
Berkeley RTG
July 31st, 2013
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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Introduction
k is a field. Let G denote the following graph.
D
v1
v4
C
A
v2
v3
B
Define the homomorphism φG : k[A, B, C , D] → k[v1 , v2 , v3 , v4 ] by
A 7→ v1 v2
B 7→ v2 v3
C 7→ v3 v4
D 7→ v4 v1 .
Im(φG ) = k[v1 v2 , v2 v3 , v3 v4 , v4 v1 ].
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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Introduction (cont.)
v1
D
v4
C
A
v2
v3
B
φG (AC − BD) = 0
In fact, it is easy to show that ker (φG ) = hAC − BDi.
We call this ideal, ker (φG ), the Toric Ideal of the graph G .
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Graph Ideals
Definition
Let G = (V , E ) be an undirected graph. We can define the map
φG : k[E ] → k[V ] by eij 7→ vi vj as shown below. The ideal ker(φG ) is
called the Toric Ideal associated to G and is denoted IG .
vj
vi
eij
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An Even Cycle
v8
H
v7
I
F
v1
v6
A
E
v2
v5
B
D
v3
C
v4
φG (ACEH − BDFI ) = v1 v2 v3 v4 v5 v6 v7 v8 − v2 v3 v4 v5 v6 v7 v8 v1 = 0
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An Odd Cycle
Let’s try to find a non-zero element
in the toric ideal
v1
AB · · · − C · · ·
A
v2
B
C
v3
For this graph, the toric ideal is...
is..
hm..
trivial? What happened here?
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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Things We Want To Be True That Are!
Proposition
IG is a homogeneous ideal generated by the following binomials which
correspond to closed even walks of the graph:
Bw =
k
Y
e2j−1 −
j=1
k
Y
e2j
j=1
Closed even walk:
e2k−2
e2k−1
e
w : w1 −−−−→ w2 −−−−→ ... −−−−→ w2k−1 −−−−→ w2k −−−2k−→ w1
wi ∈ V , ei ∈ E
e1
e2
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
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Minimal Generating Set
Definition
If hs1 , ..., sn i = I , then we say it is a minimal generating set if no subset
of it generates I .
Notation:
MG - any minimal generating set
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
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An Example: The Complete Graph
D
v1
E
| MG |= 2
F
A
v2
v4
C
B
v3
AC − BD
−(AC − EF )
EF − BD
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
hAC − BD, AC − EF , BD − EF i
seems like a more complete
description of the toric ideal.
The set
{AC −BD, AC −EF , BD −EF }
forms what is called a Graver
Basis for the ideal IG .
July 31st, 2013
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Graver Bases
Notation: GG - Graver Basis
Definition
+
−
A binomial x v − x v ∈ IG is called primitive if @ another binomial
+
−
+
+
−
−
x u − x u ∈ IG s.t. x u |x v and x u |x v .
The set of all primitive binomials in IG is called its Graver basis and
is denoted GG .
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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Fact!
Fact: MG ⊂ GG
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
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BIG QUESTION:
For what graphs G is MG = GG ?
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
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An Example of When MG 6= GG
A
B
H
C
J
F
E
D
MG = {AJF − BEH, BD − CE }
GG = {AJF − BEH, BD − CE , CAJF − DB 2 H, DAJF − CE 2 H}
Note: CAJF − DB 2 H = BH(CE − BD) + C (AJF − BEH)
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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MG and GG Can Be Vastly Different Sizes
MG = 70, but GG = 3360
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Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
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An Example Where MG = GG
(Minimal) closed even walks are all
cycles, corresponding to
A
B
p1 = AD − BC
C
D
p2 = CF − DE
E
F
p3 = AF − BE
but pi ∈
/ hpj , pk i
for {i, j, k} = {1, 2, 3}
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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A Few More Examples Where MG = GG
|GG | = 3
|GG | = 3
|GG | = 6
|GG | = 3
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Contructing Larger Graphs Where MG = GG From Smaller
Ones
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Universal Gröbner Basis
Another kind of basis with some algorithmically nice properities is
called a Universal Gröbner Basis, denoted UG , and this set lays
nestled in between the minimal generating set of a toric ideal and the
Graver Basis
Proposition
MG ⊂ UG ⊂ GG for any minimal generating set MG
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Some Results
Theorem
MG = UG ⇔ MG = GG . If IG sastisfies either of these conditions we call
it robust.
Theorem
IG is robust ⇔ (Graph theoretic conditions).
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals
July 31st, 2013
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Some Results
Theorem
MG = UG ⇔ MG = GG . If IG sastisfies either of these conditions we call
it robust.
Theorem
IG is robust ⇔ (Graph theoretic conditions).
Thank You!
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References
Sturmfels, Bernd. Gröbner Bases and Convex Polytopes. American
Mathematical Society: Providence, RI, 1991.
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