7 research outputs found
Decompositions of complete uniform hypergraphs into Hamilton Berge cycles
In 1973 Bermond, Germa, Heydemann and Sotteau conjectured that if divides
, then the complete -uniform hypergraph on vertices has a
decomposition into Hamilton Berge cycles. Here a Berge cycle consists of an
alternating sequence of distinct vertices and
distinct edges so that each contains and . So the
divisibility condition is clearly necessary. In this note, we prove that the
conjecture holds whenever and . Our argument is based on
the Kruskal-Katona theorem. The case when was already solved by Verrall,
building on results of Bermond
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
Hamilton cycles in graphs and hypergraphs: an extremal perspective
As one of the most fundamental and well-known NP-complete problems, the
Hamilton cycle problem has been the subject of intensive research. Recent
developments in the area have highlighted the crucial role played by the
notions of expansion and quasi-randomness. These concepts and other recent
techniques have led to the solution of several long-standing problems in the
area. New aspects have also emerged, such as resilience, robustness and the
study of Hamilton cycles in hypergraphs. We survey these developments and
highlight open problems, with an emphasis on extremal and probabilistic
approaches.Comment: to appear in the Proceedings of the ICM 2014; due to given page
limits, this final version is slightly shorter than the previous arxiv
versio