The Tutte Polynomial

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The Tutte Polynomial
Graph Polynomials 238900 winter 05/06
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The Rank Generation Polynomial
Reminder
Number of connected components in G(V,F)
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The Rank Generation Polynomial
Theorem 2
In addition, S(Enx,y) 1 for G(V,E) where Vn
and E0
Is the graph obtained by omitting edge e
Is the graph obtained by contracting edge e
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The Rank Generation Polynomial
Theorem 2 Proof
Let us define G G e G G /
e rltGgt rltGgt nltGgt nltGgt rltGgt rltGgt
nltGgt nltGgt
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The Rank Generation Polynomial
Theorem 2 Proof
Let us denote
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The Rank Generation Polynomial
Observations
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The Rank Generation Polynomial
Observations
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The Rank Generation Polynomial
Observations
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The Rank Generation Polynomial
Observations
3
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The Rank Generation Polynomial
Observations
1
2
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The Rank Generation Polynomial
Reminder
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The Rank Generation Polynomial
Let us define
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The Rank Generation Polynomial
1
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The Rank Generation Polynomial
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The Rank Generation Polynomial
3
2
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The Rank Generation Polynomial
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The Rank Generation Polynomial

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The Rank Generation Polynomial
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The Rank Generation Polynomial
Obvious
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The Rank Generation Polynomial
Obvious
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The Rank Generation Polynomial
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The Rank Generation Polynomial
Obvious
Q.E.D
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The Tutte Polynomial
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The Tutte Polynomial
Reminder
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5
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The Universal Tutte Polynomial
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The Universal Tutte Polynomial
Theorem 6
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The Universal Tutte Polynomial
Theorem 6 - Proof
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The Universal Tutte Polynomial
If e is a bridge
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The Universal Tutte Polynomial
If e is a bridge
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The Universal Tutte Polynomial
If e is a loop
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The Universal Tutte Polynomial
If e is a bridge
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The Universal Tutte Polynomial
If e is neither a loop nor a bridge
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The Universal Tutte Polynomial
If e is neither a loop nor a bridge
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The Universal Tutte Polynomial
If e is neither a loop nor a bridge
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The Universal Tutte Polynomial
If e is neither a loop nor a bridge
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The Universal Tutte Polynomial
Theorem 6 - Proof
Q.E.D
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The Universal Tutte Polynomial
Theorem 7
then
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The Universal Tutte Polynomial
Theorem 7 proof
V(G) is dependant entirely on V(G-e) and
V(G/e). In addition
Q.E.D
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Evaluations of the Tutte Polynomial
Proposition 8
Let G be a connected graph.
is the number of spanning trees of G
Shown in the previous lecture
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Evaluations of the Tutte Polynomial
Proposition 8
Let G be a connected graph.
is the number of connected
spanning sub-graphs of G
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Evaluations of the Tutte Polynomial
Proposition 8
Let G be a connected graph.
is the number of connected
spanning sub-graphs of G
sub-graphs of G
connected
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Evaluations of the Tutte Polynomial
Proposition 8
Let G be a connected graph.
is the number of (edge sets forming)
spanning forests of G
Spanning forests
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Evaluations of the Tutte Polynomial
Proposition 8
Let G be a connected graph.
is the number of spanning
sub-graphs of G
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The Universal Tutte Polynomial
Theorem 9
The chromatic polynomial and the Tutte
polynomial are related by the equation
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The Universal Tutte Polynomial
Theorem 9 proof
Claim
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The Universal Tutte Polynomial
Theorem 9 proof
We must show that
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The Universal Tutte Polynomial
Theorem 9 proof
and
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The Universal Tutte Polynomial
Theorem 9 proof
Obvious. (empty graphs)
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The Universal Tutte Polynomial
Theorem 9 proof
Obvious
Chromatic polynomial property
Chromatic polynomial property
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The Universal Tutte Polynomial
Theorem 9 proof
Thus
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The Universal Tutte Polynomial
Theorem 9 proof
And so, due to the uniqueness we shown in Theorem
7
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The Universal Tutte Polynomial
Theorem 9 proof
Remembering that
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The Universal Tutte Polynomial
Theorem 9 proof
Remembering that
Q.E.D
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Evaluations of the Tutte Polynomial
Let G be a connected graph.
is the number of acyclic orientations
of G. We know that (Theorem 9)
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Evaluations of the Tutte Polynomial
So
And thus
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Acyclic Orientations of Graphs
Let G be a connected graph without loops or
multiple edges.
An orientation of a graph is received
after assigning a direction to each edge.
An orientation of a graph is acyclic if it
does not contain any directed cycles.
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Acyclic Orientations of Graphs
Proposition 1.1
is the number of pairs where
is a map
and is

• an orientation of G, subject to the following
• The orientation is acyclic

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Acyclic Orientations of Graphs
Proof
The second condition forces the map to be
aproper coloring. The second condition is
immediately impliedfrom the first one.
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Acyclic Orientations of Graphs
Proof
Conversely, if the map is proper, than thesecond
condition defines a unique acyclicorientation of
G. Hence, the number of allowed mappings
issimply the number of proper coloring with
xcolors, which is by definition
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Acyclic Orientations of Graphs
be the number of pairs where
is a map
and is

• an orientation of G, subject to the following
• The orientation is acyclic

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Acyclic Orientations of Graphs
Theorem 1.2
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Acyclic Orientations of Graphs
Proof
The chromatic polynomial is uniquelydetermined
by the following
G0 is the one vertex graph
Disjoint union
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Acyclic Orientations of Graphs
Proof
We now have to show for the new polynomial
Obvious
G0 is the one vertex graph
Disjoint union
Obvious
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Acyclic Orientations of Graphs
Proof
We need to show that
Let
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Acyclic Orientations of Graphs
Proof
Let
Let be an acyclic orientation of
G-ecompatible with
Let
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Acyclic Orientations of Graphs
Proof
Let be an orientation of G after addingu ?
v to
Let be an orientation of G after addingv ?
u to
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Acyclic Orientations of Graphs
Proof
We will show that for each pair exactly one
of the orientations is acyclic and
compatible with ,expect forof them, in
which case both are acyclic
orientations compatible with
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Acyclic Orientations of Graphs
Proof
Once this is done, we will know that due to
the definition of
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Acyclic Orientations of Graphs
Proof
For each pair where - and is
an acyclic orientation compatiblewith one
of these three scenarios musthold
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Acyclic Orientations of Graphs
Proof
Case 1
Clearly is not compatible with
whileis compatible. Moreover, is acyclic

Impossible cicle
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Acyclic Orientations of Graphs
Proof
Case 2
Clearly is not compatible with
whileis compatible. Moreover, is acyclic

Impossible cicle
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Acyclic Orientations of Graphs
Proof
Case 3
Both are compatible with At least one is
also acyclic. Suppose not, then
contains
contains
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Acyclic Orientations of Graphs
Proof
contains
contains
contains
Impossible cicle
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Acyclic Orientations of Graphs
Proof
We now have to show that both andare
acyclic for exactly pairs of
with
Let z denote the vertex identifying u,v in
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Acyclic Orientations of Graphs
Proof
Z
some acyclic orientation compatible with
u
v
impossible to add a circle by the new edge u,v
two acyclic orientations, compatible with
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Acyclic Orientations of Graphs
Proof
Z
exactly one, necessarily acyclic, compatible with
u
v
Some two acyclic orientations, compatible with
All other vertices of G remains the same
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Acyclic Orientations of Graphs
Proof
And so both and are acyclic for
exactly pairs of
with And so
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Acyclic Orientations of Graphs
Proof
It is obvious that for x 1 every orientationis
compatible with And so the expression count the
number ofacyclic orientations in G
Q.E.D