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 1 / 21
Introduction
k is a field. Let G denote the following graph.
v2 v3
v1 v4
A
B
C
D
Define the homomorphism φG : k[A,B,C ,D]→ k[v1, v2, v3, v4] by
A 7→ v1v2 B 7→ v2v3 C 7→ v3v4 D 7→ v4v1.
Im(φG ) = k[v1v2, v2v3, v3v4, v4v1].
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 2 / 21
Introduction (cont.)
v2 v3
v1 v4
A
B
C
D
φG (AC − BD) = 0
In fact, it is easy to show that ker(φG ) = 〈AC − BD〉.We call this ideal, ker(φG ), the Toric Ideal of the graph G .
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 3 / 21
Graph Ideals
Definition
Let G = (V ,E ) be an undirected graph. We can define the mapφG : k[E ]→ k[V ] by eij 7→ vivj as shown below. The ideal ker(φG ) iscalled the Toric Ideal associated to G and is denoted IG .
vi vjeij
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 4 / 21
An Even Cycle
A
B
C
D
E
F
H
I
v1
v2
v3 v4
v5
v6
v7v8
φG (ACEH − BDFI ) = v1v2v3v4v5v6v7v8 − v2v3v4v5v6v7v8v1 = 0
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 5 / 21
An Odd Cycle
A B
C
v1
v2 v3
Let’s try to find a non-zero elementin the toric ideal
AB · · · − C · · ·
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 6 / 21
Things We Want To Be True That Are!
Proposition
IG is a homogeneous ideal generated by the following binomials whichcorrespond to closed even walks of the graph:
Bw =k∏
j=1
e2j−1 −k∏
j=1
e2j
Closed even walk:
w : w1e1−−−−→ w2
e2−−−−→ ...e2k−2−−−−→ w2k−1
e2k−1−−−−→ w2ke2k−−−−→ w1
wi ∈ V , ei ∈ E
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 7 / 21
Minimal Generating Set
Definition
If 〈s1, ..., sn〉 = I , then we say it is a minimal generating set if no subsetof it generates I .
Notation:MG - any minimal generating set
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 8 / 21
An Example: The Complete Graph
A
B
C
D
E F
v1
v2 v3
v4
AC − BD−(AC − EF )
EF − BD
| MG |= 2
〈AC−BD,AC−EF ,BD−EF 〉seems like a more completedescription of the toric ideal.
The set{AC−BD,AC−EF ,BD−EF}forms what is called a GraverBasis for the ideal IG .
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 9 / 21
Graver Bases
Notation: GG - Graver Basis
Definition
A binomial xv+ − xv− ∈ IG is called primitive if @ another binomialxu+ − xu− ∈ IG s.t. xu+ |xv+
and xu− |xv−.
The set of all primitive binomials in IG is called its Graver basis andis denoted GG .
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 10 / 21
Fact!
Fact: MG ⊂ GG
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 11 / 21
BIG QUESTION:
For what graphs G is MG = GG?
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 12 / 21
An Example of When MG 6= GG
D
CA B
H J
F E
MG = {AJF − BEH,BD − CE}GG = {AJF − BEH,BD − CE ,CAJF − DB2H,DAJF − CE 2H}
Note: CAJF − DB2H = BH(CE − BD) + C (AJF − BEH)
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 13 / 21
MG and GG Can Be Vastly Different Sizes
MG = 70, but GG = 3360
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 14 / 21
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 15 / 21
An Example Where MG = GG
A
E F
B
C D
(Minimal) closed even walks are allcycles, corresponding to
p1 = AD − BC
p2 = CF − DE
p3 = AF − BE
but pi /∈ 〈pj , pk〉
for {i , j , k} = {1, 2, 3}
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 16 / 21
A Few More Examples Where MG = GG
|GG | = 3
|GG | = 6
|GG | = 3
|GG | = 3
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 17 / 21
Contructing Larger Graphs Where MG = GG From SmallerOnes
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 18 / 21
Universal Grobner Basis
Another kind of basis with some algorithmically nice properities iscalled a Universal Grobner Basis, denoted UG , and this set laysnestled in between the minimal generating set of a toric ideal and theGraver Basis
Proposition
MG ⊂ UG ⊂ GG for any minimal generating set MG
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 19 / 21
Some Results
Theorem
MG = UG ⇔MG = GG . If IG sastisfies either of these conditions we callit robust.
Theorem
IG is robust ⇔ (Graph theoretic conditions).
Thank You!
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 20 / 21
Some Results
Theorem
MG = UG ⇔MG = GG . If IG sastisfies either of these conditions we callit robust.
Theorem
IG is robust ⇔ (Graph theoretic conditions).
Thank You!
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 20 / 21
References
Sturmfels, Bernd. Grobner Bases and Convex Polytopes. AmericanMathematical Society: Providence, RI, 1991.
Bryan Brown, Laura Lyman, Amy Nesky (Berkeley RTG) Toric Graph Ideals July 31st, 2013 21 / 21
Top Related