Spanning trees of 3-uniform hypergraphs

TL;DR

This paper explores the complexity and properties of spanning trees in 3-uniform hypergraphs, introducing classes of 3-Pfaffian hypergraphs, characterizations, and bounds on spanning trees.

Contribution

It characterizes 3-Pfaffian hypergraphs, provides recognition complexity results, and establishes bounds on spanning trees in Steiner triple systems.

Findings

Recognition of 3-Pfaffian 3-graphs is complex.

Identifies classes of 3-Pfaffian hypergraphs with forbidden subgraph characterizations.

Provides bounds on the number of spanning trees in Steiner triple systems.

Abstract

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.