238 research outputs found
The minimum vertex degree for an almost-spanning tight cycle in a -uniform hypergraph
We prove that any -uniform hypergraph whose minimum vertex degree is at
least admits an almost-spanning
tight cycle, that is, a tight cycle leaving vertices uncovered. The
bound on the vertex degree is asymptotically best possible. Our proof uses the
hypergraph regularity method, and in particular a recent version of the
hypergraph regularity lemma proved by Allen, B\"ottcher, Cooley and Mycroft.Comment: 10 pages. arXiv admin note: text overlap with arXiv:1411.495
Perfect packings with complete graphs minus an edge
Let K_r^- denote the graph obtained from K_r by deleting one edge. We show
that for every integer r\ge 4 there exists an integer n_0=n_0(r) such that
every graph G whose order n\ge n_0 is divisible by r and whose minimum degree
is at least (1-1/chi_{cr}(K_r^-))n contains a perfect K_r^- packing, i.e. a
collection of disjoint copies of K_r^- which covers all vertices of G. Here
chi_{cr}(K_r^-)=r(r-2)/(r-1) is the critical chromatic number of K_r^-. The
bound on the minimum degree is best possible and confirms a conjecture of
Kawarabayashi for large n
Embeddings and Ramsey numbers of sparse k-uniform hypergraphs
Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of
graphs of bounded maximum degree are linear in their order. In previous work,
we proved the same result for 3-uniform hypergraphs. Here we extend this result
to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the
main new tool which we prove and use is an embedding lemma for k-uniform
hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random'
hypergraphs.Comment: 24 pages, 2 figures. To appear in Combinatoric
Jigsaw percolation on random hypergraphs
The jigsaw percolation process on graphs was introduced by Brummitt,
Chatterjee, Dey, and Sivakoff as a model of collaborative solutions of puzzles
in social networks. Percolation in this process may be viewed as the joint
connectedness of two graphs on a common vertex set. Our aim is to extend a
result of Bollob\'as, Riordan, Slivken, and Smith concerning this process to
hypergraphs for a variety of possible definitions of connectedness. In
particular, we determine the asymptotic order of the critical threshold
probability for percolation when both hypergraphs are chosen binomially at
random.Comment: 17 page
- …