Even Orientations and Pfaffian graphs

We give a characterization of Pfaffian graphs in terms of even orientations, extending the characterization of near bipartite non--pfaffian graphs by Fischer and Little \cite{FL}. Our graph theoretical characterization is equivalent to the one proved by Little in \cite{L73} (cf. \cite{LR}) using linear algebra arguments

Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

Permanents, Pfaffian orientations, and even directed circuits

Given a 0-1 square matrix A, when can some of the 1's be changed to -1's in such a way that the permanent of A equals the determinant of the modified matrix? When does a real square matrix have the property that every real matrix with the same sign pattern (that is, the corresponding entries either have the same sign or are both zero) is nonsingular? When is a hypergraph with n vertices and n hyperedges minimally nonbipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have no directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit? It is known that all of the above problems are equivalent. We prove a structural characterization of the feasible instances, which implies a polynomial-time algorithm to solve all of the above problems. The structural characterization says, roughly speaking, that a bipartite graph has a Pfaffian orientation if and only if it can be obtained by piecing together (in a specified way) planar bipartite graphs and one sporadic nonplanar bipartite graph.Comment: 47 pages, published versio

Pfaffian orientation and enumeration of perfect matchings for some Cartesian products of graphs

The importance of Pfaffian orientations stems from the fact that if a graph G is Pfaffian, then the number of perfect matchings of G (as well as other related problems) can be computed in polynomial time. Although there are many equivalent conditions for the existence of a Pfaffian orientation of a graph, this property is not well-characterized. The problem is that no polynomial algorithm is known for checking whether or not a given orientation of a graph is Pfaffian. Similarly, we do not know whether this property of an undirected graph that it has a Pfaffian orientation is in NP. It is well known that the enumeration problem of perfect matchings for general graphs is NP-hard. L. Lovasz pointed out that it makes sense not only to seek good upper and lower bounds of the number of perfect matchings for general graphs, but also to seek special classes for which the problem can be solved exactly. For a simple graph G and a cycle C(n) with n vertices (or a path P(n) with n vertices), we define C(n) (or P(n)) x G as the Cartesian product of graphs C(n) (or P(n)) and G. In the present paper, we construct Pfaffian orientations of graphs C(4) x G, P(4) x G and P(3) x G, where G is a non bipartite graph with a unique cycle, and obtain the explicit formulas in terms of eigenvalues of the skew adjacency matrix of (G) over right arrow to enumerate their perfect matchings by Pfaffian approach, where (G) over right arrow is an arbitrary orientation of G