Towards a Characterisation of Pfaffian graphs

Towards a characterisation of Pfaffian near bipartite graphs

A Characterisation of Pfaffian Near Bipartite Graphs

A graph is 1-extendible if every edge has a 1-factor containing it. A 1-extendible non-bipartite graph G is said to be near bipartite if there exist edges e1 and e2 such that G − {e1, e2} is 1-extendible and bipartite. We characterise the Pfaffian near bipartite graphs in terms of forbidden subgraphs. The theorem extends an earlier characterisation of Pfaffian bipartite graphs.

Matching signatures and Pfaffian graphs

We prove that every 4-Pfaffian that is not Pfaffian essentially has a unique signature matrix. We also give a simple composition Theorem of 2r-Pfaffian graphs from r Pfaffian spanning subgraphs. We apply these results and exhibit a graph that is 6-Pfaffian but not 4-Pfaffian. This is a counter-example to a conjecture of Norine [5], which states that if a graph G is k-Pfaffian but not (k − 1)-Pf...

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 the 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 a matching minor and conjecture that there are onl...

Even Circuits of Prescribed Clockwise Parity

We show that a graph has an orientation under which every circuit of even length is clockwise odd if and only if the graph contains no subgraph which is, after the contraction of at most one circuit of odd length, an even subdivision of K2,3. In fact we give a more general characterisation of graphs that have an orientation under which every even circuit has a prescribed clockwise parity. This ...