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 .

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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

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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

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Contructing Larger Graphs Where MG = GG From SmallerOnes

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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.

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