Binary nullity, Euler circuits and interlace polynomials
Traldi, Lorenzo
A theorem of Cohn and Lempel [M. Cohn, A. Lempel, Cycle decomposition by disjoint transpositions, J. Combin. Theory Set. 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.
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