On the number of dissimilar pfaffian orientations of graphs
, Campo Grande,
Brasil and University of Waterloo, Waterloo, Canada.
2 , Campinas, Brasil;email@example.com
3 University of Waterloo, Waterloo, Canada.
A subgraph H of a graph G is conformal if G - V(H) has a perfect matching. An orientation D of G is Pfaffian if, for every conformal even circuit C, the number of edges of C whose directions in D agree with any prescribed sense of orientation of C is odd. A graph is Pfaffian if it has a Pfaffian orientation. Not every graph is Pfaffian. However, if G has a Pfaffian orientation D, then the determinant of the adjacency matrix of D is the square of the number of perfect matchings of G. (See the book by Lovász and Plummer [Matching Theory. Annals of Discrete Mathematics, vol. 9. Elsevier Science (1986), Chap. 8.] A matching covered graph is a nontrivial connected graph in which every edge is in some perfect matching. The study of Pfaffian orientations of graphs can be naturally reduced to matching covered graphs. The properties of matching covered graphs are thus helpful in understanding Pfaffian orientations of graphs. For example, say that two orientations of a graph are similar if one can be obtained from the other by reversing the orientations of all the edges in a cut of the graph. Using one of the theorems we proved in [M.H. de Carvalho, C.L. Lucchesi and U.S.R. Murty, Optimal ear decompositions of matching covered graphs. J. Combinat. Theory B 85 (2002) 59–93] concerning optimal ear decompositions, we show that if a matching covered graph is Pfaffian then the number of dissimilar Pfaffian orientations of G is 2b(G), where b(G) is the number of “bricks” of G. In particular, any two Pfaffian orientations of a bipartite graph are similar. We deduce that the problem of determining whether or not a graph is Pfaffian is as difficult as the problem of determining whether or not a given orientation is Pfaffian, a result first proved by Vazirani and Yanakakis [Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs. Discrete Appl. Math. 25 (1989) 179–180]. We establish a simple property of minimal graphs without a Pfaffian orientation and use it to give an alternative proof of the characterization of Pfaffian bipartite graphs due to Little [ A characterization of convertible (0,1)-matrices. J. Combinat. Theory B 18 (1975) 187–208] .
Mathematics Subject Classification: 05C70
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