Spanning trees of 3-uniform hypergraphs

Masbaum and Vaintrob's "Pfaffian matrix tree theorem" implies that counting spanning trees of a 3-uniform hypergraph (abbreviated to 3-graph) can be done in polynomial time for a class of "3-Pfaffian" 3-graphs, comparable to and related to the class of Pfaffian graphs. We prove a complexity result for recognizing a 3-Pfaffian 3-graph and describe two large classes of 3-Pfaffian 3-graphs -- one of these is given by a forbidden subgraph characterization analogous to Little's for bipartite Pfaffian graphs, and the other consists of a class of partial Steiner triple systems for which the property of being 3-Pfaffian can be reduced to the property of an associated graph being Pfaffian. We exhibit an infinite set of partial Steiner triple systems that are not 3-Pfaffian, none of which can be reduced to any other by deletion or contraction of triples. We also find some necessary or sufficient conditions for the existence of a spanning tree of a 3-graph (much more succinct than can be obtained by the currently fastest polynomial-time algorithm of Gabow and Stallmann for finding a spanning tree) and a superexponential lower bound on the number of spanning trees of a Steiner triple system.Comment: 34 pages, 9 figure

Odd-Cycle-Free Facet Complexes and the K\"onig property

We use the definition of a simplicial cycle to define an odd-cycle-free facet complex (hypergraph). These are facet complexes that do not contain any cycles of odd length. We show that besides one class of such facet complexes, all of them satisfy the K\"onig property. This new family of complexes includes the family of balanced hypergraphs, which are known to satisfy the K\"onig property. These facet complexes are, however, not Mengerian; we give an example to demonstrate this fact.Comment: 12 pages, 11 figure

Dimers, Tilings and Trees

Generalizing results of Temperley, Brooks, Smith, Stone and Tutte and others we describe a natural equivalence between three planar objects: weighted bipartite planar graphs; planar Markov chains; and tilings with convex polygons. This equivalence provides a measure-preserving bijection between dimer coverings of a weighted bipartite planar graph and spanning trees on the corresponding Markov chain. The tilings correspond to harmonic functions on the Markov chain and to ``discrete analytic functions'' on the bipartite graph. The equivalence is extended to infinite periodic graphs, and we classify the resulting ``almost periodic'' tilings and harmonic functions.Comment: 23 pages, 5 figure