48,981 research outputs found
Cyclic cycle systems of the complete multipartite graph
In this paper, we study the existence problem for cyclic -cycle
decompositions of the graph , the complete multipartite graph with
parts of size , and give necessary and sufficient conditions for their
existence in the case that
A problem on partial sums in abelian groups
In this paper we propose a conjecture concerning partial sums of an arbitrary
finite subset of an abelian group, that naturally arises investigating simple
Heffter systems. Then, we show its connection with related open problems and we
present some results about the validity of these conjectures
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
Finitely Correlated Representations of Product Systems of -Correspondences over
We study isometric representations of product systems of correspondences over
the semigroup which are minimal dilations of finite dimensional,
fully coisometric representations. We show the existence of a unique minimal
cyclic coinvariant subspace for all such representations. The compression of
the representation to this subspace is shown to be complete unitary invariant.
For a certain class of graph algebras the nonself-adjoint \textsc{wot}-closed
algebra generated by these representations is shown to contain the projection
onto the minimal cyclic coinvariant subspace. This class includes free
semigroup algebras. This result extends to a class of higher-rank graph
algebras which includes higher-rank graphs with a single vertex.Comment: 34 pages; Introduction extended; to appear in the Journal of
Functional Analysi
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