Diapositiva 1

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The Tutte PolynomialofRegular Tilings
D. Garijo, A. Márquez, P. Revuelta Dpto.
Matemática Aplicada I Universidad de Sevilla
(Spain)
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• Definition of Matroid (Whitney-1935)
• A matroid M is a finite set S and a collection I
  of subsets of M such that
• ? ? I
• If X ? I and Y ? X, then Y ? I
• If U,V ?I with UV1, there exists x ? U-V
  such that V?x ? I

• Examples
• Vector spaces
• Graphs

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• Brylawskis idea
• f(M) is a Tutte-Grothendieck invariant if
• f(M)f(N) if M and N are isomorphic
• f(M)f(M-e)f(M/e) if e is not loop or coloop
• f(M?N)f(M)f(N)

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The Tutte Polynomial
Defined on graphs by Tutte and Whitney
(1954) Extension of the Chromatic Polynomial
Applicability in Combinatorics, other areas of
Mathematics and in the field of Physics
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Combinatorial Definition of the Tutte Polynomial

Let be a graph, . The
rank of a subset A is defined where
is the number of connected components of the
spanning subgraph


r(A)8-53
Some Properties of the Rank
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Combinatorial Definition of the Tutte Polynomial

The Tutte Polynomial of a graph
is defined as
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Relation with Rank-Size-Generating Polynomial
The Rank-Size-Generating Polynomial of a graph G
is defined as



The coefficient of is the number of
spanning subgraphs in G with rank i and j
edges.
Both contain the same information about any graph
The coefficient of of the Tutte
Polynomial is the number of edge-sets in G with
rank i and size j
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The Tutte Plane




Chromatic Polynomial
T(G, 1, 1) number of spanning trees T(G, 2, 1)
number of forests T(G, 1, 2) number of sets
that contains a base T(G, 2, 0) number of
acyclic orientations or number of arragement of
hyperplanes T(G, 0, 2) number of totally
cyclic orientations T(G, x, 1/x) related with
the Jones polynomial of a link Brylawski Oxley
(1992)
Flow Polynomial

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Recursive Definition of the Tutte Polynomial


Delete and Contract Edges




delete the edge e
contract the edge e
The order in which the edges are chosen to be
deleted and contracted is not important
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In general, to compute an evaluation of the Tutte
polynomial is NP-hard.
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Problem (Read Tutte-1988) Find a general
formula for the chromatic polynomial of the graph
associated with a chessboard.
An easy question to ask, but without doubt a
fiendishly difficult one to answer.
Marín, Márquez Revuelta (2004)
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Proposition The Tutte polynomial associated to
L3,n with n ? 2 is obtained from





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• Applicable to any dimension.
• We have computed up to 12xn in squared grid,
  but using also trinagles and hexagons (and
  combinations of them).

Calkin, Merino, Noble, Noy (2003) Squared grid
7x7 Sokal, Shrock (2001-2003) Chromatic
polynomial using squares and/or triangles up to
9xn
In the grid 12xn dim(M) es 56

Compute 3112 polynomials in two variables
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Grid 5x3




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Grid 5x4




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Are two graphs with the same Tutte Polynomial
isomorphic? NO

G is Tutte Unique if for every other graph H
such that T(G,x,y)T(H,x,y) implies that G is
isomorphic to H
PROBLEM Find large families of Tutte Unique
graphs
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Tutte Uniqueness of Locally grid graphs

LOCALLY GRID GRAPHS
Let N(x) be the set of the neighbours of a
vertex x. A 4-regular connected graph G is a
locally grid graph if for each vertex x of G
there is an ordering x1, x2, x3, x4 of N(x) and
four different vertices y1, y2, y3, y4 such
that, taking the indices modulo 4 and there
are no more adjacencies among x1, x2, x3, x4 ,
y1, y2, y3, y4 than those required by these
conditions



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Tutte Uniqueness


PROBLEM Prove that locally grid graphs are
uniquely determined by their Tutte polynomial

Show that, given two locally grid graphs, there
is at least one coefficient of the Tutte
polynomial in which both graphs differ
Lemma 1 (Márquez, Mier, Noy, Revuelta
2003) Given two graphs G1 and G2, if G1 is a
locally grid graph and T(G1 x, y) T(G2 x,
y) then G2 is a locally grid graph
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Classification Theorem (Márquez, Mier, Noy,
Revuelta 2003) If G is a locally grid graph with
Npq vertices, then exactly one of
the following holds

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Edge-Sets

Essential Cycles

Given a locally grid graph G and two cycles C1
and C2 of G , C1 is locally homotopic to C2 if
there is a cycle H of length 4, such that
is connected and C2 is obtained from C1 by
replacing with


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A homotopy is a sequence of local homotopies
A cycle C of a locally grid graph G is an
essential cycle if it is not homotopic to a
cycle of length 4

Minimum length of an essential cycle of G
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Lemma 3 (Garijo, Márquez, Revuelta 2004) If G
is a locally grid graph with pq vertices, then
the length of the shortest essential cycles and
the number of these cycles are given in the
following table




Theorem (Garijo, Márquez, Revuelta 2004) The
locally grid graphs are Tutte unique

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Theorem (Garijo, Márquez, Revuelta 2004) The
locally grid graphs are Tutte unique
120 comparisons

Pairwise compare all the graphs given in the
classification theorem of locally grid graphs

• Equal number of vertices
• Equal minimum length of essential cycles
• Equal number of shortest essential cycles

In 105 cases, we prove that the previous
quantities are different
In 15 cases, we show that although the previous
quantities are equal, the number of edge-sets
of certain rank and size are not equal in both
graphs

At least one coefficient of the Tutte
polynomial of each graph is different
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Tutte Uniqueness of Hexagonal Tilings

Lemma 1 (Garijo, Márquez, Revuelta, 2004) Given
two graph G and H, if H is a hexagonal tiling
and T(G x, y) T( H x, y) then G is a
hexagonal tiling

Lemma 2 (Garijo, Márquez, Revuelta, 2004) The
hexagonal tilings are locally orientable graphs
Lemma 3 (Garijo, Márquez, Revuelta, 2004) For all
ngt0, the number of edge-sets that do not contain
essential cycles of rank n-1 and size n is
equal for every hexagonal tiling H with 2pq
vertices and such that n lt lH1

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No contain essential cycles

Edge-Set

Contain essential cycles

• Lemma 4 (Garijo, Márquez, Revuelta, 2004)
• Let H1 and H2 be two hexagonal tilings with 2pq
  vertices.
• Then
• If and H1, H2 do not have the same
  number of
• shortest essential cycles, then

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Theorem (Garijo, Márquez, Revuelta, 2004) The
toroidal hexagonal tiling Hk,m,0 with mgt1 y kgt2
is Tutte unique

Number of vertices Minimun length of essential
cycles Number of shortest essential
cycles Chromatic number


54 comparisons

• 9 cases where the chromatic number is different
• 39 cases where the chromatic number is equal and
  the other quantities
• are different
• 6 cases that although the previous quantities
  are equal, the number
• of edge-sets of certain rank and size are not
  equal in both graphs

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Gracias