# Princeton Discrete Math Seminar

## Title:

Permanents, Pfaffian orientations, and even directed circuits

## Speaker:

- Robin Thomas
- Georgia Institute of Technology

## Place:

- Fine Hall 214
- Princeton University

## Time:

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

Abstract:
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