The Interlace Polynomial : a New Graph Polynomialrichard Arratia

LIMITED DISTRIBUTION NOTICE: This report has been submitted for publication outside of IBM and will probably be copyrighted if accepted for publication. It has been issued as a Research Report for early dissemination of its contents. In view of the transfer of copyright to the outside publisher, its distribution outside of IBM prior to publication should be limited to peer communications and speciic requests. After outside publication, requests should be lled only by reprints or legally obtained copies of the article (e.g., payment of royalties). Copies may be requested from IBM T. Abstract. We deene a new graph polynomial, the interlace polynomial, for any undirected graph. One of our main results is that the polynomial, speciied by an intricate recursion relation, is well-deened; in the absence of a direct interpretation of the polynomial, this remains mysterious. The interlace polynomial is not a special case of the Tutte polynomial, nor do we know of any other graph polynomial to which it can be reduced. For 2-in, 2-out directed graphs D, any Euler circuit induces an undirected \interlace" graph H. In this setting, the interlace polynomial q(H) is equal to the Martin polynomial m(D), a variant of the circuit partition polynomial. There is another connection, between the Kauuman brackets of a link diagram, the Martin polynomial, and in turn the interlace polynomial. We explore other properties of the interlace polynomial, such as its relations with the component number and independence number of a graph, its extremal values, and its values for various classes of graphs. A pair of intriguing conjectures remain unproved. More importantly, the fundamental \meaning" of the polynomial is not yet understood. Contents 1. Introduction 1 2. The pivot operator on graphs 3 3. Counting circuits in 2-in, 2-out digraphs 5 4. Pivoting about an edge 6 5. A new graph polynomial 6 5.1. Deenition and well-deenedness 7 5.2. Simple properties 9 6. Circuit decompositions and the Martin and interlace polynomials 10 7. Kauuman brackets and the Martin and interlace polynomials 13 8. Interlace polynomials of some simple graphs 15 9. Remarks on interlacement and the interlace polynomial 16 10. Interlace polynomials of substituted and rotated graphs 17 10.1. Substituted graphs 17 10.2. Rotated graphs 20 11. Extremal properties of the interlace polynomial 21 12. Open problems 23 Acknowledgments 24 References 24