Spanning trees of 3-uniform hypergraphs

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.

Constructions of Snake-in-the-Box Codes for Rank Modulation

“…This step will be performed by translating the merging problem into a 3graph problem. We start with a sequence of definitions taken from [7].…”

“…The size of the K-snake C 2n+1 depends on the number of vertices in the tree T 2n+1 . A tree in a 3-graph contains an odd number of vertices [7]. Since in H 2n+1 there are (2n + 1)(2n) vertices it follows that there is no tree in H 2n+1 which contains all the vertices of V 2n+1 .…”

“…(3) vertices [4, 1], [2,4], (through the edge 1, 2, 4 ); (4) vertices [5, 1], [2,5], (through the edge 1, 2, 5 ); (5) vertices [5,3], [1, 5], (through the edge 1, 5, 3 ); (6) vertices [5,2], [3,5], (through the edge 2, 3, 5 ); (7) vertices [3,4], [1,3], (through the edge 1, 3, 4 ); (8) vertices [3,2], [4,3], (through the edge 2, 4, 3 ); (9) vertices [4,5], [1,4], (through the edge 1, 4, 5 ); (10) vertices [4,2], [5,4], (through the edge 2, 5, 4 ).…”

“…The chains inT 7 : . Note that the label on the edge is a necklace which can merge together the chains of its corresponding endpoints by M [x]-connection.…”

The Bounds of the Edge Number in Generalized Hypertrees

“…This concept of the generalization of a tree to hypergraphs has been introduced and studied in the literature, which can be regarded as a continuation of the classical generalization effort. For more details, see [8,10,11].…”