6,578 research outputs found
Resolution of the Oberwolfach problem
The Oberwolfach problem, posed by Ringel in 1967, asks for a decomposition of
into edge-disjoint copies of a given -factor. We show that this
can be achieved for all large . We actually prove a significantly more
general result, which allows for decompositions into more general types of
factors. In particular, this also resolves the Hamilton-Waterloo problem for
large .Comment: 28 page
Long path and cycle decompositions of even hypercubes
We consider edge decompositions of the -dimensional hypercube into
isomorphic copies of a given graph . While a number of results are known
about decomposing into graphs from various classes, the simplest cases of
paths and cycles of a given length are far from being understood. A conjecture
of Erde asserts that if is even, and divides the number
of edges of , then the path of length decomposes . Tapadia et
al.\ proved that any path of length , where , satisfying these
conditions decomposes . Here, we make progress toward resolving Erde's
conjecture by showing that cycles of certain lengths up to
decompose . As a consequence, we show that can be decomposed into
copies of any path of length at most dividing the number of edges of
, thereby settling Erde's conjecture up to a linear factor
On Hamilton Decompositions of Line Graphs of Non-Hamiltonian Graphs and Graphs without Separating Transitions
In contrast with Kotzig's result that the line graph of a -regular graph
is Hamilton decomposable if and only if is Hamiltonian, we show that
for each integer there exists a simple non-Hamiltonian -regular
graph whose line graph has a Hamilton decomposition. We also answer a question
of Jackson by showing that for each integer there exists a simple
connected -regular graph with no separating transitions whose line graph has
no Hamilton decomposition
- …