Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-26T15:13:13.700Z Has data issue: false hasContentIssue false
  • English
  • Français

On the number of dissimilar pfaffian orientations of graphs

Published online by Cambridge University Press:  15 March 2005

Marcelo H. de Carvalho ,
Cláudio L. Lucchesi  and
U. S.R. Murty
Marcelo H. de Carvalho
Affiliation:
, Campo Grande, Brasil and University of Waterloo, Waterloo, Canada.
Cláudio L. Lucchesi
Affiliation:
, Campinas, Brasil;[email protected]
U. S.R. Murty
Affiliation:
University of Waterloo, Waterloo, Canada.
Get access

Abstract

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 B85 (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 B18 (1975) 187–208] .

Keywords

Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 39 , Issue 1: Imre Simon, the tropical computer scientist , January 2005 , pp. 93 - 113
Copyright
© EDP Sciences, 2005

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

de Carvalho, M.H., Lucchesi, C.L. and Murty, U.S.R., The perfect matching polytope and solid bricks. J. Combin. Theory B 92 (2004) 319–324. CrossRef
de Carvalho, M.H., Lucchesi, C.L. and Murty, U.S.R., Ear decompositions of matching covered graphs. Combinatorica 19 (1999) 151–174. CrossRef
de Carvalho, M.H., Lucchesi, C.L. and Murty, U.S.R., On a conjecture of Lovász concerning bricks. I. The characteristic of a matching covered graph. J. Comb. Theory B 85 (2002) 94–136. CrossRef
de Carvalho, M.H., Lucchesi, C.L. and Murty, U.S.R., On a conjecture of Lovász concerning bricks. II. Bricks of finite characteristic. J. Comb. Theory B 85 (2002) 137–180. CrossRef
de Carvalho, M.H., Lucchesi, C.L. and Murty, U.S.R., Optimal ear decompositions of matching covered graphs. J. Comb. Theory B 85 (2002) 59–93. CrossRef
Edmonds, J., Lovász, L. and PulleyblanK, W.R., Brick decomposition and the matching rank of graphs. Combinatorica 2 (1982) 247–274. CrossRef
Fischer, I. and Little, C.H.C., A characterisation of Pfaffian near bipartite graphs. J. Comb. Theory B 82 (2001) 175–222. CrossRef
Kasteleyn, P.W., Dimer statistics and phase transitions. J. Math. Phys. 4 (1963) 287–293. CrossRef
Little, C., A characterization of convertible (0,1)-matrices. J. Comb. Theory B 18 (1975) 187–208. CrossRef
Little, C.H.C. and Rendl, F., Operations preserving the Pfaffian property of a graph. J. Austral. Math. Soc. Ser. A 50 (1991) 248–275. CrossRef
Lovász, L., Matching structure and the matching lattice. J. Comb. Theory B 43 (1987) 187–222. CrossRef
L. Lovász and M.D. Plummer, Matching Theory. Annals of Discrete Mathematics, vol. 29. Elsevier Science (1986).
McCuaig, W., Brace generation. J. Graph Theory 38 (2001) 124–169. CrossRef
Robertson, N., Seymour, P.D. and Thomas, R., Permanents, Pfaffian orientations and even directed circuits. Ann. Math. 150 (1999) 929–975. CrossRef
W.T. Tutte, Graph Theory as I Have Known It. Number 11 in Oxford Lecture Ser. Math. Appl. Clarendon Press, Oxford (1998).
Vazirani, V.V. and Yanakakis, M., Pfaffian orientation of graphs, 0,1 permanents, and even cycles in digraphs. Discrete Appl. Math. 25 (1989) 179–180. CrossRef