Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with bipartite case, as Little [7] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K3,3. We relax the notion of matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion. We define Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize minimally non-Pfaffian factorcritical graphs.

• Type: Journal
• Year: 2008
• Where: JCT
• Authors: Serguei Norine, Robin Thomas
• Keywords: Bipartite Non-Pfaffian Graph, JCT 2008, Non-pfaffian Graphs, Pfaffian Factor-critical Graph