7 research outputs found
Cyclic Matching Sequencibility of Graphs
We define the cyclic matching sequencibility of a graph to be the largest
integer such that there exists a cyclic ordering of its edges so that every
consecutive edges in the cyclic ordering form a matching. We show that the
cyclic matching sequencibility of and equal
Permutations that separate close elements, and rectangle packings in the torus
Let , and be positive integers. For distinct ,
define to be the distance between and when the elements of
are written in a circle. So A permutation
is \emph{-clash-free} if whenever
. So an -clash-free permutation moves every pair of close
elements (at distance less than ) to a pair of elements at large distance
(at distance at least ). The notion of an -clash-free permutation can
be reformulated in terms of certain packings of rectangles on an
torus.
For integers and with , let be the largest
value of such that an -clash-free permutation of
exists. Strengthening a recent paper of Blackburn, which proved a conjecture of
Mammoliti and Simpson, we determine the value of in all cases.Comment: 21 pages, 6 figure
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