632 research outputs found
The cyclic coloring complex of a complete k-uniform hypergraph
In this paper, we study the homology of the cyclic coloring complex of three
different types of -uniform hypergraphs. For the case of a complete
-uniform hypergraph, we show that the dimension of the
homology group is given by a binomial coefficient. Further, we discuss a
complex whose -faces consist of all ordered set partitions where none of the contain a hyperedge of the complete
-uniform hypergraph and where . It is shown that the
dimensions of the homology groups of this complex are given by binomial
coefficients. As a consequence, this result gives the dimensions of the
multilinear parts of the cyclic homology groups of \C[x_1,...,x_n]/
\{x_{i_1}...x_{i_k} \mid i_{1}...i_{k} is a hyperedge of . For the other
two types of hypergraphs, star hypergraphs and diagonal hypergraphs, we show
that the dimensions of the homology groups of their cyclic coloring complexes
are given by binomial coefficients as well
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
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
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