20 research outputs found
Rainbow matchings in bipartite multigraphs
Suppose that is a non-negative integer and a bipartite multigraph is
the union of matchings
, each of size . We show that has a rainbow matching of
size , i.e. a matching of size with all edges coming from different
's. Several choices of parameters relate to known results and conjectures
An approximate version of a conjecture of Aharoni and Berger
Aharoni and Berger conjectured that in every proper edge-colouring of a bipartite
multigraph by n colours with at least n+1 edges of each colour there is a rainbow matching
using every colour. This conjecture generalizes a longstanding problem of Brualdi and
Stein about transversals in Latin squares. Here an approximate version of the AharoniBerger Conjecture is proved—it is shown that if there are at least n + o(n) edges of each
colour in a proper n-edge-colouring of a bipartite multigraph then there is a rainbow
matching using every colour
Multipartite hypergraphs achieving equality in Ryser's conjecture
A famous conjecture of Ryser is that in an -partite hypergraph the
covering number is at most times the matching number. If true, this is
known to be sharp for for which there exists a projective plane of order
. We show that the conjecture, if true, is also sharp for the smallest
previously open value, namely . For , we find the minimal
number of edges in an intersecting -partite hypergraph that has
covering number at least . We find that is achieved only by linear
hypergraphs for , but that this is not the case for . We
also improve the general lower bound on , showing that .
We show that a stronger form of Ryser's conjecture that was used to prove the
case fails for all . We also prove a fractional version of the
following stronger form of Ryser's conjecture: in an -partite hypergraph
there exists a set of size at most , contained either in one side of
the hypergraph or in an edge, whose removal reduces the matching number by 1.Comment: Minor revisions after referee feedbac
Matrices of zeros and ones with fixed row and column sum vectors
AbstractLet m and n be positive integers, and let R=(r1,…,rm) and S=(s1,…,sn) be nonnegative integral vectors. We survey the combinational properties of the set of all m × n matrices of 0's and 1's having ri1's in row i andsi 1's in column j. A number of new results are proved. The results can be also be formulated in terms of a set of bipartite graps with bipartition into m and n vertices having degree sequence R and S, respectively. They can also be formulated in terms of the set of hypergraphs with m vertices having degree sequence R and n edges whose cardinalities are given by S