8 research outputs found
A note on intersecting hypergraphs with large cover number
Published by 'The Electronic Journal of Combinatorics' at 10.37236/6460.We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r−4 for all but finitely many r. This answers a question of Abu-Khazneh, Barát, Pokrovskiy and Szabó, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r.Natural Sciences and Engineering Research Council of Canad
Non-intersecting Ryser hypergraphs
A famous conjecture of Ryser states that every -partite hypergraph has
vertex cover number at most times the matching number. In recent years,
hypergraphs meeting this conjectured bound, known as -Ryser hypergraphs,
have been studied extensively. It was recently proved by Haxell, Narins and
Szab\'{o} that all -Ryser hypergraphs with matching number are
essentially obtained by taking disjoint copies of intersecting -Ryser
hypergraphs. Abu-Khazneh showed that such a characterisation is false for by giving a computer generated example of a -Ryser hypergraph with whose vertex set cannot be partitioned into two sets such that we have an
intersecting -Ryser hypergraph on each of these parts. Here we construct new
infinite families of -Ryser hypergraphs, for any given matching number , that do not contain two vertex disjoint intersecting -Ryser
subhypergraphs.Comment: 8 pages, some corrections in the proof of Lemma 3.6, added more
explanation in the appendix, and other minor change
A family of extremal hypergraphs for Ryser's conjecture
Ryser's Conjecture states that for any -partite -uniform hypergraph,
the vertex cover number is at most times the matching number. This
conjecture is only known to be true for in general and for
if the hypergraph is intersecting. There has also been considerable effort made
for finding hypergraphs that are extremal for Ryser's Conjecture, i.e.
-partite hypergraphs whose cover number is times its matching number.
Aside from a few sporadic examples, the set of uniformities for which
Ryser's Conjecture is known to be tight is limited to those integers for which
a projective plane of order exists.
We produce a new infinite family of -uniform hypergraphs extremal to
Ryser's Conjecture, which exists whenever a projective plane of order
exists. Our construction is flexible enough to produce a large number of
non-isomorphic extremal hypergraphs. In particular, we define what we call the
{\em Ryser poset} of extremal intersecting -partite -uniform hypergraphs
and show that the number of maximal and minimal elements is exponential in
.
This provides further evidence for the difficulty of Ryser's Conjecture
Covering graphs by monochromatic trees and Helly-type results for hypergraphs
How many monochromatic paths, cycles or general trees does one need to cover
all vertices of a given -edge-coloured graph ? These problems were
introduced in the 1960s and were intensively studied by various researchers
over the last 50 years. In this paper, we establish a connection between this
problem and the following natural Helly-type question in hypergraphs. Roughly
speaking, this question asks for the maximum number of vertices needed to cover
all the edges of a hypergraph if it is known that any collection of a few
edges of has a small cover. We obtain quite accurate bounds for the
hypergraph problem and use them to give some unexpected answers to several
questions about covering graphs by monochromatic trees raised and studied by
Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao,
Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi
A family of extremal hypergraphs for Ryser's conjecture
Ryser's Conjecture states that for any r-partite r-uniform hypergraph, the vertex cover number is at most r−1 times the matching number. This conjecture is only known to be true for r≤3 in general and for r≤5 if the hypergraph is intersecting. There has also been considerable effort made for finding hypergraphs that are extremal for Ryser's Conjecture, i.e. r-partite hypergraphs whose cover number is r−1 times its matching number. Aside from a few sporadic examples, the set of uniformities r for which Ryser's Conjecture is known to be tight is limited to those integers for which a projective plane of order r−1 exists. We produce a new infinite family of r-uniform hypergraphs extremal to Ryser's Conjecture, which exists whenever a projective plane of order r−2 exists. Our construction is flexible enough to produce a large number of non-isomorphic extremal hypergraphs. In particular, we define what we call the Ryser poset of extremal intersecting r-partite r-uniform hypergraphs and show that the number of maximal and minimal elements is exponential in r. This provides further evidence for the difficulty of Ryser's Conjecture
A note on intersecting hypergraphs with large cover number
We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r − 4 for all but finitely many r. This answers a question of Abu-Khazneh, Barát, Pokrovskiy and Szabó, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r
A note on intersecting hypergraphs with large cover number
We give a construction of r-partite r-uniform intersecting hypergraphs with cover number at least r − 4 for all but finitely many r. This answers a question of Abu-Khazneh, Barát, Pokrovskiy and Szabó, and shows that a long-standing unsolved conjecture due to Ryser is close to being best possible for every value of r