A generalization of Little's Theorem on Pfaffian orientations

  • Authors:

  • Marcelo H. De Carvalho;Cláudio L. Lucchesi;U. S. R. Murty

  • Affiliations:

  • facom-ufms, Campo Grande, Brazil;facom-ufms, Campo Grande, Brazil;University of Waterloo, Waterloo, Canada

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2012

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Abstract

Little (1975) [12] showed that, in a certain sense, the only minimal non-Pfaffian bipartite matching covered graph is the brace K"3","3. Using a stronger notion of minimality than the one used by Little, we show that every minimal non-Pfaffian brick G contains two disjoint odd cycles C"1 and C"2 such that the subgraph G-V(C"1@?C"2) has a perfect matching. This implies that the only minimal non-Pfaffian solid matching covered graph is the brace K"3","3. (A matching covered graph G is solid if, for any two disjoint odd cycles C"1 and C"2 of G, the subgraph G-V(C"1@?C"2) has no perfect matching. Solid matching covered graphs constitute a natural generalization of the class of bipartite graphs, see Carvalho et al., 2004 [5].)