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

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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

+

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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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