On the Convexity of the Tutte Polynomial of a Paving ...merino/online_papers/ACCOTA2010.pdf · Criel Merino Convexity of the Tutte Polynomial of a Paving Matroid. Basic concepts Inequalities

Basic conceptsInequalitiesConvexity

On the Convexity of the Tutte Polynomial of aPaving Matroid Along Line Segments

Laura Chavez Lomelı† C.Merino‡ Steven D. Noble§

Marcelino Ramırez-Ibanez‡

†UAM-Azcapotzalco

§Departament of Mathematical SciencesBrunel University

‡Instituto de Matematicas UNAM,sede Oaxaca

ACCOTA 2010

Criel Merino Convexity of the Tutte Polynomial of a Paving Matroid


Summary

1 Basic conceptsMatroidsMatroid constructionsTutte Polynomial

2 InequalitiesMerino-Welsh conjectureFirst result

3 ConvexityMain resultPaving Matroidstechnical resultsProof



MatroidsMatroid constructionsTutte Polynomial

Definition

A matroid M is a pair (E , r), where E is a finite set andr : ℘(E ) → N, the rank function, is such that

1 0 ≤ r(A) ≤ |A|, for all A ⊆ E .

2 If A ⊆ B, then r(A) ≤ r(B)

3 For all A,B ⊆ E ,

r(A ∪ B) + r(A ∩ B) ≤ r(A) + r(B) (submodularity)




Definition


1 0 ≤ r(A) ≤ |A|, for all A ⊆ E .







Definition


1 0 ≤ r(A) ≤ |A|, for all A ⊆ E .







Bases and circuits

Let M = (E , r) be a matroid,

A ⊆ E is an independent set if |A| = r(A).

B ⊆ E is a basis if B is a maximal independent set.

C ⊆ E is a circuit if it is a minimal dependent set.




Bases and circuits








Bases and circuits








Bases and circuits








Example: Graphic matroids

For a connected graph G = (V ,E ) we define the matroidM(G ) = (E , r), where r(A) is the size of a maximal spanningforest in (V ,A).

An independent set in M(G ) is just a spanning forest of G .

A base is a spanning tree of G .

A circuit is a cycle of G .












































Example: Uniform matroids

Um,n = ([n], r), where [n] = {1, 2 . . . , n} and the rank function isgiven by

r(A) = min{|A|,m}, whereA ⊆ [n].

An independent set is a any set with cardinality at most m. Abasis is a set of cardinality m and a circuit is any set of cardinalitym + 1.





1 2 3 4 5

12 45

E U3,5

∅

123 345








1 2 3 4 5

12 45

E U3,5

∅

123 345



An independent set is a any set with cardinality at most m.

Abasis is a set of cardinality m and a circuit is any set of cardinalitym + 1.





1 2 3 4 5

12 45

E U3,5

∅

123 345



An independent set is a any set with cardinality at most m. Abasis is a set of cardinality m

and a circuit is any set of cardinalitym + 1.





1 2 3 4 5

12 45

E U3,5

∅

123 345







Duality

If M = (E , r) is a matroid, then we can define its dual matroidM∗ = (E , r∗) where r∗(A) = |A| − r(E ) + r(E − A).

Figure: M(G ) and its dual M(G )∗




Duality

If M = (E , r) is a matroid, then we can define its dual matroidM∗ = (E , r∗) where r∗(A) = |A| − r(E ) + r(E − A).

Figure: M(G ) and its dual M(G )∗




Deletion-contraction, loops and coloops

Let M = (E , r) be a matroid and e ∈ E . The matroid obtained bydeleting e is defined by M\e=(E\e, r |E\e). The matroid obtainedby contracting e is defined by M/e=(M∗\e)∗.

An element e ∈ E with rank r(e) = 0 is called a loop, and duallyan element with r∗(e) = 0 is called an isthmus or coloop.




Deletion-contraction, loops and coloops

Let M = (E , r) be a matroid and e ∈ E . The matroid obtained bydeleting e is defined by M\e=(E\e, r |E\e). The matroid obtainedby contracting e is defined by M/e=(M∗\e)∗.An element e ∈ E with rank r(e) = 0 is called a loop, and duallyan element with r∗(e) = 0 is called an isthmus or coloop.




Definition

For a matroid M, the 2-variable polynomial

TM(x , y) =∑A⊆E

(x − 1)r(E)−r(A)(y − 1)|A|−r(A), (1)

is called the Tutte polynomial. It is easy to prove that

TM(1, 1) equals the number of bases of M.

TM(2, 2) = 2|E |.

TM∗(x , y) = TM(y , x).




Definition

For a matroid M, the 2-variable polynomial

TM(x , y) =∑A⊆E

(x − 1)r(E)−r(A)(y − 1)|A|−r(A), (1)

is called the Tutte polynomial. It is easy to prove that

TM(1, 1) equals the number of bases of M.

TM(2, 2) = 2|E |.

TM∗(x , y) = TM(y , x).




Linear recursion definition

If M is a matroid and e is an element that is not a loop or anisthmus, then

TM(x , y) = TM\e(x , y) + TM/e(x , y). (2)

If no such an element e exists, TM(x , y) = x iy j where i and j arethe number of isthmuses and loops respectively.



Merino-Welsh conjectureFirst result

Old Conjecture

C.Merino y D.J.A. Welsh proposed the following in 1999

Conjecture

If G is a 2-connected graph with no loops, then

max{TG (2, 0),TG (0, 2)} ≥ TG (1, 1).

TG (2, 0) equals the number of acyclic orientations while TG (0, 2)equals the number of totally cyclic orientations.




Old Conjecture


Conjecture


max{TG (2, 0),TG (0, 2)} ≥ TG (1, 1).





Old Conjecture


Conjecture


max{TG (2, 0),TG (0, 2)} ≥ TG (1, 1).





(0, 2)

(2, 0)

(1, 1)

TG(x, y)




First Result

Theorem

If a matroid M contains two disjoint bases, then

TM(0, 2a) ≥ TM(a, a), for all a ≥ 2.

Dually, if its ground set is the union of two bases, then

TM(2a, 0) ≥ TM(a, a), for all a ≥ 2.




TM(x, y)




TM(x, y)

(0, 2a)

(a, a)

(2a, 0)



Main resultPaving Matroidstechnical resultsProof

Main theorem

Theorem

If M is a coloopless paving matroid, then TM(x , y) = T satisfies

tT (x1, y1)+(1−t)T (x2, y2) ≥ T (tx1+(1−t)x2, ty1+(1−t)y2), (3)

for 0 ≤ t ≤ 1 and x1 + y1 = x2 + y2 = p; where p ≥ 0 and0 ≤ x1, x2, y1, y2.

That is, TM is convex along the portion of the line x + y = p lyingin the positive quadrant.




Main theorem

Theorem

If M is a coloopless paving matroid, then TM(x , y) = T satisfies

tT (x1, y1)+(1−t)T (x2, y2) ≥ T (tx1+(1−t)x2, ty1+(1−t)y2), (3)

for 0 ≤ t ≤ 1 and x1 + y1 = x2 + y2 = p; where p ≥ 0 and0 ≤ x1, x2, y1, y2.

That is, TM is convex along the portion of the line x + y = p lyingin the positive quadrant.




(x1, y1)

(x2, y2)

x + y = p

TM(x, y)




Definition

A paving matroid M = (E, r) is a matroid whose circuits all havesize at least r(E ).




Definition


r(Mr(M)r(Mr(M)+1)+1




Definition


Note

Paving matroids are closed under minors




Paving matroids and disjoint bases

Theorem

Let M = (E , r) be a rank-r paving matroid with n elements,

if 2r > n, then E is the union of two bases,

if 2r ≤ n and M is coloopless, then M contains two disjointbases.




Paving matroids and disjoint bases

Theorem

Let M = (E , r) be a rank-r paving matroid with n elements,

if 2r > n, then E is the union of two bases,

if 2r ≤ n and M is coloopless, then M contains two disjointbases.




Proof.

case: 2r > n

. . .

...

01

r − 1r

E

∅

...

n

2 . . .




Proof.

case: 2r > n

. . .

...

01

r − 1r

E

∅

...

n

B

2 . . .




Proof.

case: 2r > n

. . .

...

01

r − 1r

E

∅

...

n

B

I = E −B

2 . . .




Proof.

case: 2r > n

. . .

...

01

r − 1r

E

∅

...

n

B

I = E −B

I ⊂ B′

2 . . .




Duality and convexity

Lemma

Let M be a matroid. Either, both TM(x , y) and TM∗(x , y) areconvex along the portion of the line x + y = p lying in the positivequadrant or neither is.

Proof.

This follows directly from the equality TM(x , y) = TM∗(y , x).




Deletion-contraction and convexity

Lemma

Let M be a matroid and e in M be neither a loop nor a coloop. IfTM\e and TM/e are both convex along the portion of the linex + y = p lying in the positive quadrant, then TM is also convexon the same domain.

Proof.

This follows directly from linear recursive definition.




Base case

Lemma

If M is isomorphic to the paving matroid U1,k+1 ⊕ U0,l , wherel ≥ 0 and k ≥ 1, then TM is convex along the portion of the linex + y = p lying in the positive quadrant.

Proof.

TM(x , y) = py l +l+k∑

m=l+2

ym.




Base case

Lemma

If M is isomorphic to the paving matroid U1,k+1 ⊕ U0,l , wherel ≥ 0 and k ≥ 1, then TM is convex along the portion of the linex + y = p lying in the positive quadrant.

Proof.

TM(x , y) = py l +l+k∑

m=l+2

ym.




Uniform matroids and convexity

Theorem

The Tutte polynomial TM is a convex function in the positivequadrant when M is a uniform matroid. In particular, TM isconvex along the portion of the line x + y = p lying in the positivequadrant.




Proof.

The Tutte polynomial of Ur ,n can be computed using (1)

TUr,n(x , y) =r−1∑i=0

(n

i

)(x − 1)r−i +

(n

r

)+

n∑i=r+1

(n

i

)(y − 1)i−r ,

This can be expanded into the following expression,

TUr,n(x , y) =n−r∑j=1

(n − j − 1

r − 1

)y j +

r∑i=1

(n − i − 1

n − r − 1

)x i .

TUn,n(x , y) = xn y TU0,n(x , y) = yn.




Proof.


TUr,n(x , y) =r−1∑i=0

(n

i

)(x − 1)r−i +

(n

r

)+

n∑i=r+1

(n

i

)(y − 1)i−r ,



(n − j − 1

r − 1

)y j +

r∑i=1

(n − i − 1

n − r − 1

)x i .





Proof.


TUr,n(x , y) =r−1∑i=0

(n

i

)(x − 1)r−i +

(n

r

)+

n∑i=r+1

(n

i

)(y − 1)i−r ,



(n − j − 1

r − 1

)y j +

r∑i=1

(n − i − 1

n − r − 1

)x i .





rank-2 case

Theorem

If M is a rank-2 loopless and coloopless matroid, then TM isconvex along the portion of the line x + y = p lying in the positivequadrant.

Proof.




rank-2 case

Theorem


Proof.

If M ∼= U2,n, the result follows from applying the previous lemma.Else, M is isomorphic to a matroid with parallel elements whosesimplification is isomorphic to U2,n.




rank-2 case

Theorem


Proof.

If n ≥ 3 or there is a parallel class of size at least 3, there existse ∈ M such that M/e ∼= U1,k+1⊕U0,l and M\e is a rank-2 looplessand coloopless matroid. The geometric representation of M is




rank-2 case

Theorem


Proof.

Otherwise, the simplification of M is isomorphic to U2,2 and everyelement is in a parallel class of size 2. Then M ∼= U1,2 ⊕ U1,2 andTM = (x +y)2 which is convex (in fact is constant) along x +y = pfor p > 0 and 0 ≤ y ≤ p.




Structural result

In order to establish or main result, we need the followingstructural result about coloopless paving matroids.

Lemma

Let M be a rank-r coloopless paving matroid. If for every elemente of M, M \ e has a coloop, then one of the following three caseshappens.

1 M is isomorphic to Ur ,r+1.

2 M is the 2-stretching of a uniform matroid Us,s+2, for somes ≥ 1.

3 M is isomorphic to U1,2 ⊕ U1,2.




Structural result

Lemma

Let M be a rank-r coloopless paving matroid. If for every elemente of M, M \ e has a coloop, then TM is convex along the portionof the line x + y = p lying in the positive quadrant.

Proof.

We jus check the 4 cases of the previos theorem. The onlyinteresting one is when M is the 2-stretching of Us,s+2. In thiscase we use that M∗ is the 2-thickening of U2,n.




Proof of main theorem

Theorem

If M is a coloopless paving matroid, then TM is convex along theportion of the line x + y = p lying in the positive quadrant.

Proof.

M has a loop, M ∼= U1,k+1 ⊕ U0,l . (base case)

Otherwise, every element of M is neither a loop nor a coloop,

a) If there is an element e such that M \ e has no coloop, thenboth M/e and M \ e are coloopless paving matroids.(Induction)

b) So, we can assume that for all e, M \ e has acoloop.(Structural result)





Theorem


Proof.









Theorem


Proof.









Theorem


Proof.









Theorem


Proof.








Corollary

If M or M∗ is a coloopless paving matroid, then TM is convex alongthe portion of the line x + y = p lying in the positive quadrant.




The proportion of paving matroids

1976 Dominic Welsh ask if most matroids are paving, he used thetable of matroids of up to 8 elements of Blackburn, Crapo yHiggs from 1973.

2008 This table has been update recently up to 9 elements forMayhew and Royle.

2010 Mayhew, Newman, Welsh y Whittle pose as a conjecture thatasymptotically almost every matroid is paving.




Our conjecture

Conjecture

If M contains two disjoint bases or its ground set is the union oftwo bases then

max{TM(2, 0),TM(0, 2)} ≥ TM(1, 1).




Thanks




Gracias


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On the Convexity of the Tutte Polynomial of a Paving ...merino/online_papers/ACCOTA2010.pdf · Criel Merino Convexity of the Tutte Polynomial of a Paving Matroid. Basic concepts Inequalities