1,231 research outputs found
Toric algebra of hypergraphs
The edges of any hypergraph parametrize a monomial algebra called the edge
subring of the hypergraph. We study presentation ideals of these edge subrings,
and describe their generators in terms of balanced walks on hypergraphs. Our
results generalize those for the defining ideals of edge subrings of graphs,
which are well-known in the commutative algebra community, and popular in the
algebraic statistics community. One of the motivations for studying toric
ideals of hypergraphs comes from algebraic statistics, where generators of the
toric ideal give a basis for random walks on fibers of the statistical model
specified by the hypergraph. Further, understanding the structure of the
generators gives insight into the model geometry.Comment: Section 3 is new: it explains connections to log-linear models in
algebraic statistics and to combinatorial discrepancy. Section 6 (open
problems) has been moderately revise
Edge-coloring linear hypergraphs with medium-sized edges
Motivated by the Erd\H{o}s-Faber-Lov\'{a}sz (EFL) conjecture for hypergraphs,
we consider the list edge coloring of linear hypergraphs. We show that if the
hyper-edge sizes are bounded between and
inclusive, then there is a list edge coloring using colors. The dependence on in the upper bound is optimal (up to the
value of )
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