48 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
Rainbow sets in the intersection of two matroids
Given sets , a {\em partial rainbow function} is a partial
choice function of the sets . A {\em partial rainbow set} is the range of
a partial rainbow function. Aharoni and Berger \cite{AhBer} conjectured that if
and are matroids on the same ground set, and are
pairwise disjoint sets of size belonging to , then there exists a
rainbow set of size belonging to . Following an idea of
Woolbright and Brower-de Vries-Wieringa, we prove that there exists such a
rainbow set of size at least
Rainbow matchings in properly-coloured multigraphs
Aharoni and Berger conjectured that in any bipartite multigraph that is
properly edge-coloured by colours with at least edges of each
colour there must be a matching that uses each colour exactly once. In this
paper we consider the same question without the bipartiteness assumption. We
show that in any multigraph with edge multiplicities that is properly
edge-coloured by colours with at least edges of each colour
there must be a matching of size that uses each colour at most once.Comment: 7 page