1,163 research outputs found
Partition-crossing hypergraphs
For a finite set X, we say that a set H ⊆ X crosses a partition P = (X1, . . . , Xk) of X if H intersects min(|H|, k) partition classes. If |H| ≥ k, this means that H meets all classes Xi, whilst for |H| ≤ k the elements of the crossing set H belong to mutually distinct classes. A set system H crosses P, if so does some H ∈ H. The minimum number of r-element subsets, such that every k-partition of an n-element set X is crossed by at least one of them, is denoted by f(n, k, r). The problem of determining these minimum values for k = r was raised and studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc. J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp. 1387–1404]. The present authors determined asymptotically tight estimates on f(n, k, k) for every fixed k as n → ∞ [Graphs Combin., 25 (2009), 807–816]. Here we consider the more general problem for two parameters k and r, and establish lower and upper bounds for f(n, k, r). For various combinations of the three values n, k, r we obtain asymptotically tight estimates, and also point out close connections of the function f(n, k, r) to Tur´an-type extremal problems on graphs and hypergraphs, or to balanced incomplete block designs
Partition-Crossing Hypergraphs
For a finite set , we say that a set crosses a partition
of if intersects partition
classes. If , this means that meets all classes , whilst
for the elements of the crossing set belong to mutually
distinct classes. A set system crosses , if so does some
. The minimum number of -element subsets, such that every
-partition of an -element set is crossed by at least one of them, is
denoted by .
The problem of determining these minimum values for was raised and
studied by several authors, first by Sterboul in 1973 [Proc. Colloq. Math. Soc.
J. Bolyai, Vol. 10, Keszthely 1973, North-Holland/American Elsevier, 1975, pp.
1387--1404]. The present authors determined asymptotically tight estimates on
for every fixed as [Graphs Combin., 25 (2009),
807--816]. Here we consider the more general problem for two parameters and
, and establish lower and upper bounds for . For various
combinations of the three values we obtain asymptotically tight
estimates, and also point out close connections of the function to
Tur\'an-type extremal problems on graphs and hypergraphs, or to balanced
incomplete block designs.Comment: 15 page
Tur\'annical hypergraphs
This paper is motivated by the question of how global and dense restriction
sets in results from extremal combinatorics can be replaced by less global and
sparser ones. The result we consider here as an example is Turan's theorem,
which deals with graphs G=([n],E) such that no member of the restriction set
consisting of all r-tuples on [n] induces a copy of K_r.
Firstly, we examine what happens when this restriction set is replaced just
by all r-tuples touching a given m-element set. That is, we determine the
maximal number of edges in an n-vertex such that no K_r hits a given vertex
set.
Secondly, we consider sparse random restriction sets. An r-uniform hypergraph
R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if
for any graph G on [n] with more edges than the Turan number ex(n,K_r)
(respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G.
We determine the thresholds for random r-uniform hypergraphs to be Turannical
and to epsilon-Turannical.
Thirdly, we transfer this result to sparse random graphs, using techniques
recently developed by Schacht [Extremal results for random discrete structures]
to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random
graphs.Comment: 33 pages, minor improvements thanks to two referee
Counting and enumerating optimum cut sets for hypergraph -partitioning problems for fixed
We consider the problem of enumerating optimal solutions for two hypergraph
-partitioning problems -- namely, Hypergraph--Cut and
Minmax-Hypergraph--Partition. The input in hypergraph -partitioning
problems is a hypergraph with positive hyperedge costs along with a
fixed positive integer . The goal is to find a partition of into
non-empty parts -- known as a -partition -- so as
to minimize an objective of interest.
1. If the objective of interest is the maximum cut value of the parts, then
the problem is known as Minmax-Hypergraph--Partition. A subset of hyperedges
is a minmax--cut-set if it is the subset of hyperedges crossing an optimum
-partition for Minmax-Hypergraph--Partition.
2. If the objective of interest is the total cost of hyperedges crossing the
-partition, then the problem is known as Hypergraph--Cut. A subset of
hyperedges is a min--cut-set if it is the subset of hyperedges crossing an
optimum -partition for Hypergraph--Cut.
We give the first polynomial bound on the number of minmax--cut-sets and a
polynomial-time algorithm to enumerate all of them in hypergraphs for every
fixed . Our technique is strong enough to also enable an -time
deterministic algorithm to enumerate all min--cut-sets in hypergraphs, thus
improving on the previously known -time deterministic algorithm,
where is the number of vertices and is the size of the hypergraph. The
correctness analysis of our enumeration approach relies on a structural result
that is a strong and unifying generalization of known structural results for
Hypergraph--Cut and Minmax-Hypergraph--Partition. We believe that our
structural result is likely to be of independent interest in the theory of
hypergraphs (and graphs).Comment: Accepted to ICALP'22. arXiv admin note: text overlap with
arXiv:2110.1481
Hamilton cycles in 5-connected line graphs
A conjecture of Carsten Thomassen states that every 4-connected line graph is
hamiltonian. It is known that the conjecture is true for 7-connected line
graphs. We improve this by showing that any 5-connected line graph of minimum
degree at least 6 is hamiltonian. The result extends to claw-free graphs and to
Hamilton-connectedness
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