Princeton Discrete Math Seminar


Permanents, Pfaffian orientations, and even directed circuits


Robin Thomas
Georgia Institute of Technology


Fine Hall 214
Princeton University


approximately 4:45 (after the DIMACS meeting)
Thursday, March 27, 1997

Given an n by n 0-1 matrix A, when can some of the 1's be changed to -1's in such a way that the permament of A equals the determinant of the modified matrix? When does a real n by n matrix A have the property that every real matrix B with the same sign pattern (that is, the corresponding entries either have the same sign, or are both zero) is non-singular? When is a hypergraph with n vertices and n hyperedges minimally non-bipartite? When does a bipartite graph have a "Pfaffian orientation"? Given a digraph, does it have a directed circuit of even length? Given a digraph, does it have a subdivision with no even directed circuit?

It is known that all 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 non-planar bipartite graph.

This is joint work with Neil Robertson and P.D. Seymour. The structural characterization was independently obtained by W. McCuaig.

Next week: John Conway

Document last modified on March 25, 19997