925 research outputs found
Edge coloring complete uniform hypergraphs with many components
AbstractLet H be a hypergraph. For a k-edge coloring c:E(H)→{1,…,k} let f(H,c) be the number of components in the subhypergraph induced by the color class with the least number of components. Let fk(H) be the maximum possible value of f(H,c) ranging over all k-edge colorings of H. If H is the complete graph Kn then, trivially, f1(Kn)=f2(Kn)=1. In this paper we prove that for n⩾6,f3(Kn)=⌊n/6⌋+1 and supply close upper and lower bounds for fk(Kn) in case k⩾4. Several results concerning the value of fk(Knr), where Knr is the complete r-uniform hypergraph on n vertices, are also established
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
Chromatic Ramsey number of acyclic hypergraphs
Suppose that is an acyclic -uniform hypergraph, with . We
define the (-color) chromatic Ramsey number as the smallest
with the following property: if the edges of any -chromatic -uniform
hypergraph are colored with colors in any manner, there is a monochromatic
copy of . We observe that is well defined and where
is the -color Ramsey number of . We give linear upper bounds
for when T is a matching or star, proving that for , and where
and are, respectively, the -uniform matching and star with
edges.
The general bounds are improved for -uniform hypergraphs. We prove that
, extending a special case of Alon-Frankl-Lov\'asz' theorem.
We also prove that , which is sharp for . This is
a corollary of a more general result. We define as the 1-intersection
graph of , whose vertices represent hyperedges and whose edges represent
intersections of hyperedges in exactly one vertex. We prove that for any -uniform hypergraph (assuming ). The proof uses the list coloring version of Brooks' theorem.Comment: 10 page
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