84 research outputs found
Hamilton cycles in quasirandom hypergraphs
We show that, for a natural notion of quasirandomness in -uniform
hypergraphs, any quasirandom -uniform hypergraph on vertices with
constant edge density and minimum vertex degree contains a
loose Hamilton cycle. We also give a construction to show that a -uniform
hypergraph satisfying these conditions need not contain a Hamilton -cycle
if divides . The remaining values of form an interesting
open question.Comment: 18 pages. Accepted for publication in Random Structures & Algorithm
Perfect Packings in Quasirandom Hypergraphs II
For each of the notions of hypergraph quasirandomness that have been studied,
we identify a large class of hypergraphs F so that every quasirandom hypergraph
H admits a perfect F-packing. An informal statement of a special case of our
general result for 3-uniform hypergraphs is as follows. Fix an integer r >= 4
and 0<p<1. Suppose that H is an n-vertex triple system with r|n and the
following two properties:
* for every graph G with V(G)=V(H), at least p proportion of the triangles in
G are also edges of H,
* for every vertex x of H, the link graph of x is a quasirandom graph with
density at least p.
Then H has a perfect -packing. Moreover, we show that neither
hypotheses above can be weakened, so in this sense our result is tight. A
similar conclusion for this special case can be proved by Keevash's hypergraph
blowup lemma, with a slightly stronger hypothesis on H.Comment: 17 page
Euler tours in hypergraphs
We show that a quasirandom -uniform hypergraph has a tight Euler tour
subject to the necessary condition that divides all vertex degrees. The
case when is complete confirms a conjecture of Chung, Diaconis and Graham
from 1989 on the existence of universal cycles for the -subsets of an
-set.Comment: version accepted for publication in Combinatoric
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