1. The Circuit Partition Polynomial and Relation to the Tutte Polynomial aaustin2@smcvt.edu Prof. Ellis-Monaghan 1 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.

2. Orientation & Eulerian Graphs An oriented graph is a graph in which the edges are directed. An Eulerian graph is a graph whose vertices are all of even degree. The orientation of a graph is called Eulerian if the in- degree at each vertex is equal to the out-degree. 2

3. Loops and Multiple Edges A 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 3 Multiple Edges Bridge Loop

4. Eulerian Graph State/k-Partition 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. An Eulerian k-Partition is a graph state with k components. 4 Example: Consider the Eulerian 3-Partition:

5. 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: 5 = + = x

6. Eg. Let = There are 6 one component states. There are also 8 two component states and 2 three component states. Thus, A one component state A two component state A three component state Circuit Partition: Finding the Polynomial Example 6

7. Tutte Polynomial Tutte polynomial for graphs satisfies 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 7

8. Tutte Polynomial Example e Delete e Contract e + 8

9. Example… 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. Thus, the Tutte polynomial representation of G is: + + 9

10. 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. G 10

11. 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: 11 Martin, Las Vergnas

12. Circuit Partition and Tutte A Planar graph G G m with the vertex faces colored black Orient G m so that black faces are to the left of each edge. 12

13. 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. 13