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The Circuit Partition Polynomial and Relation to the Tutte Polynomial
aaustin2@smcvt.eduProf. Ellis-Monaghan
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Andrea Austin
The project described was supported by the Vermont Genetics Network through NIH Grant Number 1 P20 RR16462 from the INBRE program
of the National Center for Research Resources.
Eulerian Graph
An Eulerian graph is
a graph whose
vertices are all of
even degree.
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Loops and Multiple EdgesA loop is an edge that
connects a vertex to itself.
A bridge is an edge that connects two components of a graph. If removed, the graph would be disconnected.
A multiple edge is a pair pf vertices with more than one edge joining them.
A multigraph is a graph that may have multiple edges and/or loops.
Loop
Multigraph
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Multiple Edges
Multiple Edges
Bridge
Loop
Oriented Graph/Digraph
An oriented graph is agraph in which the edges are directed.
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Eulerian Orientation
The orientation of a graph is called Eulerian if the in-degree at each vertex is equal to the out-degree.
A simple graph with Eulerian orientation
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Eulerian Graph States
An Eulerian Graph State of a graph, G, is the result of replacing all 2n-valent vertices, v, of G, with n 2-valent vertices joining pairs of edges originally adjacent to v.
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Eulerian k-Partitions
An Eulerian k-Partition is a graph state with k components.
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Example: Consider the Eulerian 3-Partition:
Circuit Partition Polynomial
The circuit partition polynomial, , of a directed Eulerian graph, G, is given by
where is the number of Eulerian graph states of G with k components.
The polynomial is given recursively by:
);( xGj
8
0
)();(k
kk xGfxGj )(
Gfk
= +
= x
);( xGj
Medial Graph
A medial graph of a connected planar graph, G, is constructed by putting a vertex on each edge of G, and drawing edges around the faces of G.
G9
Circuit Partition Example
G mG
xxxxGj m 34; 23
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Eulerian Graph States
X3 X3
2-component states
3-component state
1-component states
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Tutte Polynomial
Tutte polynomial for graphs satisfying the following relations: G has no edges
G has an edge e that is neither a loop nor a bridge
G is made up of i bridges and j loops
( ; , ) 1T G x y
( ; , ) ( ( / ; , )) ( ; , )T G x y T G e x y T G e x y
( ; , ) i jT G x y x y
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Tutte Polynomial Example
e
Delete e Contract e
+
13
G
+
Example…
2x
e
We delete edge e and are left with a bridge, or x.
We contract on edge e and are left with a loop, or y.
e
Thus, the Tutte polynomial representation of G is:2
2
( ; , )
( ; , ) 2
T G x y x x y
T G x x x x
+ +
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Circuit Partition and Tutte
Tutte Eulerian circuits If G is a planar graph and is the oriented
medial graph then the Tutte polynomial encodes information about the numbers of Euler circuits in
Use the formula:
mG
mG
( )( ; ) ( ; 1, 1)c Gmj G x x T G x x
��������������
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Martin, Las Vergnas
Circuit Partition vs. TutteA Planar graph G Gm with the vertex
faces colored red
Orient Gm so that red faces are to the left of each edge.
xxxxxxxxGxTxGj m 34)]1(2)1[(1,1;; 2321
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= +
Recursive Formulas:
= +
= x
= 1
Recall: Circuit Partition Polynomial Recursive Formula
Tutte Polynomial Recursive Formula
delete contract
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Circuit Partition-Tutte Connection
Connection via the medial graph:
delete contract
e= +
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Sources
Ellis-Monaghan, Joanna. Exploring the Tutte-Martin connection, Discrete Mathematics, 281, no 1-3 (2004) 173-187.
Ellis-Monaghan, Joanna. Generalized transition polynomials (with I. Sarmiento), Congressus Numerantium 155 (2002) 57-69.
Ellis-Monaghan, Joanna. Identities for the circuit partition polynomials, with applications to the diagonal Tutte polynomial, Advances in Applied Mathematics, 32 no. 1-2, (2004) 188-197.
Ellis-Monaghan, Joanna. Martin polynomial miscellanea. Congressus Numerantium 137 (1999), 19–31.
Ellis-Monaghan, Joanna. New results for the Martin polynomial. Journal of Combinatorial Theory, series B 74 (1998), 326–52.
M. Las Vergnas, On Eulerian partitions of graphs, Graph Theory and Combinatorics, Proceedings of Conference, Open University, Milton Keynes, 1978, Research Notes in Mathematics, Vol. 34, Pitman,Boston, MA, London, 1979, pp. 62–75.
M. Las Vergnas, On the evaluation at (3,3) of the Tutte polynomial of a graph, J. Combin. Theory,Ser. B 44 (1988) 367–372.
P. Martin, Enumerations euleriennes dans le multigraphs et invariants de Tutte Grothendieck, Thesis, Grenoble, 1977.
P. Martin, Remarkable valuation of the dichromatic polynomial of planar multigraphs, J. Combin.Theory, Ser. B 24 (1978) 318–324.
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