2,477 research outputs found
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
Switching codes and designs
AbstractVarious local transformations of combinatorial structures (codes, designs, and related structures) that leave the basic parameters unaltered are here unified under the principle of switching. The purpose of the study is threefold: presentation of the switching principle, unification of earlier results (including a new result for covering codes), and applying switching exhaustively to some common structures with small parameters
Join-irreducible Boolean functions
This paper is a contribution to the study of a quasi-order on the set
of Boolean functions, the \emph{simple minor} quasi-order. We look at
the join-irreducible members of the resulting poset . Using a
two-way correspondence between Boolean functions and hypergraphs,
join-irreducibility translates into a combinatorial property of hypergraphs. We
observe that among Steiner systems, those which yield join-irreducible members
of are the -2-monomorphic Steiner systems. We also describe
the graphs which correspond to join-irreducible members of .Comment: The current manuscript constitutes an extension to the paper
"Irreducible Boolean Functions" (arXiv:0801.2939v1
New Steiner 2-designs from old ones by paramodifications
Techniques of producing new combinatorial structures from old ones are commonly called trades. The switching principle applies for a broad class of designs: it is a local transformation that modifies two columns of the incidence matrix. In this paper, we present a construction, which is a generalization of the switching transform for the class of Steiner 2-designs. We call this construction paramodification of Steiner 2-designs, since it modifies the parallelism of a subsystem. We study in more detail the paramodifications of affine planes, Steiner triple systems, and abstract unitals. Computational results show that paramodification can construct many new unitals
Switching for Small Strongly Regular Graphs
We provide an abundance of strongly regular graphs (SRGs) for certain
parameters with . For this we use Godsil-McKay
(GM) switching with a partition of type and Wang-Qiu-Hu (WQH) switching
with a partition of type . In most cases, we start with a highly
symmetric graph which belongs to a finite geometry. To our knowledge, most of
the obtained graphs are new.
For all graphs, we provide statistics about the size of the automorphism
group. We also find the recently discovered Kr\v{c}adinac partial geometry,
therefore finding a third method of constructing it.Comment: 15 page
Properties of Steiner triple systems of order 21
Properties of the 62,336,617 Steiner triple systems of order 21 with a
non-trivial automorphism group are examined. In particular, there are 28 which
have no parallel class, six that are 4-chromatic, five that are 3-balanced, 20
that avoid the mitre, 21 that avoid the crown, one that avoids the hexagon and
two that avoid the prism. All systems contain the grid. None have a block
intersection graph that is 3-existentially closed.Comment: 12 page
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