Binary nullity, Euler circuits and interlace polynomials

TL;DR

This paper extends a classical equality relating circuit partitions and nullity in directed graphs to undirected 4-regular graphs, connecting it with interlace polynomials and topological concepts from knot theory.

Contribution

It generalizes the Cohn-Lempel equality to undirected 4-regular graphs and links interlace polynomials with circuit partitions and topological invariants.

Findings

Extended equality describes arbitrary circuit partitions in 4-regular graphs.

Connects interlace polynomials with circuit partition generating functions.

Incorporates topological results from knot theory.

Abstract

A theorem of Cohn and Lempel [J. Combin. Theory Ser. A 13 (1972), 83-89] gives an equality relating the number of circuits in a directed circuit partition of a 2-in, 2-out digraph to the GF(2)-nullity of an associated matrix. This equality is essentially equivalent to the relationship between directed circuit partitions of 2-in, 2-out digraphs and vertex-nullity interlace polynomials of interlace graphs. We present an extension of the Cohn-Lempel equality that describes arbitrary circuit partitions in (undirected) 4-regular graphs. The extended equality incorporates topological results that have been of use in knot theory, and it implies that if H is obtained from an interlace graph by attaching loops at some vertices then the vertex-nullity interlace polynomial $q_{N}(H)$ is essentially the generating function for certain circuit partitions of an associated 4-regular graph.