14 research outputs found
Polynomial-time perfect matchings in dense hypergraphs
Let be a -graph on vertices, with minimum codegree at least for some fixed . In this paper we construct a polynomial-time
algorithm which finds either a perfect matching in or a certificate that
none exists. This essentially solves a problem of Karpi\'nski, Ruci\'nski and
Szyma\'nska; Szyma\'nska previously showed that this problem is NP-hard for a
minimum codegree of . Our algorithm relies on a theoretical result of
independent interest, in which we characterise any such hypergraph with no
perfect matching using a family of lattice-based constructions.Comment: 64 pages. Update includes minor revisions. To appear in Advances in
Mathematic
Embedding spanning bipartite graphs of small bandwidth
Boettcher, Schacht and Taraz gave a condition on the minimum degree of a
graph G on n vertices that ensures G contains every r-chromatic graph H on n
vertices of bounded degree and of bandwidth o(n), thereby proving a conjecture
of Bollobas and Komlos. We strengthen this result in the case when H is
bipartite. Indeed, we give an essentially best-possible condition on the degree
sequence of a graph G on n vertices that forces G to contain every bipartite
graph H on n vertices of bounded degree and of bandwidth o(n). This also
implies an Ore-type result. In fact, we prove a much stronger result where the
condition on G is relaxed to a certain robust expansion property. Our result
also confirms the bipartite case of a conjecture of Balogh, Kostochka and
Treglown concerning the degree sequence of a graph which forces a perfect
H-packing.Comment: 23 pages, file updated, to appear in Combinatorics, Probability and
Computin