The signature matrix for 6-Pfaffian graphs

Abstract

Given a graph G, an orientation D of G and a perfect matching M of G, it is possible to define the sign (-1 or +1) of M in D. Given k orientations of a graph G, a signature matrix A has rows corresponding to perfect matchings of G and columns corresponding to each of these k orientations such that each entry aij is the sign of the i-th perfect matching on the j-th orientation. A graph is k-Pfaffian if there is a set of k orientations whose signature matrix A is such that the linear system Ax = 1 has a solution. The Pfaffian number of a graph G is the smallest k such that G is k-Pfaffian. We present in this paper a characterization of the signature matrices of graphs with Pfaffian number 6.