176 research outputs found
Completing Partial Packings of Bipartite Graphs
Given a bipartite graph and an integer , let be the smallest
integer such that, any set of edge disjoint copies of on vertices, can
be extended to an -design on at most vertices. We establish tight
bounds for the growth of as . In particular, we
prove the conjecture of F\"uredi and Lehel \cite{FuLe} that .
This settles a long-standing open problem
A Multipartite Hajnal-Szemer\'edi Theorem
The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree
threshold that forces a graph to contain a perfect K_k-packing. Fischer's
conjecture states that the analogous result holds for all multipartite graphs
except for those formed by a single construction. Recently, we deduced an
approximate version of this conjecture from new results on perfect matchings in
hypergraphs. In this paper, we apply a stability analysis to the extremal cases
of this argument, thus showing that the exact conjecture holds for any
sufficiently large graph.Comment: Final version, accepted to appear in JCTB. 43 pages, 2 figure
Counting packings of generic subsets in finite groups
A packing of subsets in a group is a
sequence such that are
disjoint subsets of . We give a formula for the number of packings if the
group is finite and if the subsets satisfy
a genericity condition. This formula can be seen as a generalization of the
falling factorials which encode the number of packings in the case where all
the sets are singletons
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