On the optimality of the Arf invariant formula for graph polynomials

We prove optimality of the Arf invariant formula for the generating function of even subgraphs, or, equivalently, the Ising partition function, of a graph.Comment: Advances in Mathematics, 201

Matching signatures and Pfaffian graphs

Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Perfect matchings of k-Pfaffian graphs may be enumerated in polynomial time on the number of vertices, for fixed k. In general, this enumeration problem is #P-complete. We give a Composition Theorem of 2r-Pfaffian graphs from r Pfaffian spanning subgraphs. Constructions of k-Pfaffian graphs known prior to this seem to be of a very different and essentially topological nature. We apply our Composition Theorem to produce a bipartite graph on 10 vertices that is 6-Pfaffian but not 4-Pfaffian. This is a counter-example to a conjecture of Norine (2009) [8], which states that the Pfaffian number of a graph is a power of four. (C) 2010 Elsevier B.V. All rights reserved.3114289294Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq