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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
Hamiltonicity in connected regular graphs
In 1980, Jackson proved that every 2-connected -regular graph with at most
vertices is Hamiltonian. This result has been extended in several papers.
In this note, we determine the minimum number of vertices in a connected
-regular graph that is not Hamiltonian, and we also solve the analogous
problem for Hamiltonian paths. Further, we characterize the smallest connected
-regular graphs without a Hamiltonian cycle.Comment: 5 page
The number of Hamiltonian decompositions of regular graphs
A Hamilton cycle in a graph is a cycle passing through every vertex
of . A Hamiltonian decomposition of is a partition of its edge
set into disjoint Hamilton cycles. One of the oldest results in graph theory is
Walecki's theorem from the 19th century, showing that a complete graph on
an odd number of vertices has a Hamiltonian decomposition. This result was
recently greatly extended by K\"{u}hn and Osthus. They proved that every
-regular -vertex graph with even degree for some fixed
has a Hamiltonian decomposition, provided is sufficiently
large. In this paper we address the natural question of estimating ,
the number of such decompositions of . Our main result is that
. In particular, the number of Hamiltonian
decompositions of is
Hamiltonicity and -hypergraphs
We define and study a special type of hypergraph. A -hypergraph ), where is a partition of , is an
-uniform hypergraph having vertices partitioned into classes of
vertices each. If the classes are denoted by , ,...,, then a
subset of of size is an edge if the partition of formed by
the non-zero cardinalities , ,
is . The non-empty intersections are called the parts
of , and denotes the number of parts. We consider various types
of cycles in hypergraphs such as Berge cycles and sharp cycles in which only
consecutive edges have a nonempty intersection. We show that most
-hypergraphs contain a Hamiltonian Berge cycle and that, for and , a -hypergraph always contains a sharp
Hamiltonian cycle. We also extend this result to -intersecting cycles
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