977 research outputs found
Monochromatic cycle power partitions
Improving our earlier result we show that for every integer k≥1 there exists a c(k) such that in every 2-colored complete graph apart from at most c(k) vertices the vertex set can be covered by 200k2logk vertex disjoint monochromatic kth powers of cycles. © 2016 Elsevier B.V
Vertex covers by monochromatic pieces - A survey of results and problems
This survey is devoted to problems and results concerning covering the
vertices of edge colored graphs or hypergraphs with monochromatic paths, cycles
and other objects. It is an expanded version of the talk with the same title at
the Seventh Cracow Conference on Graph Theory, held in Rytro in September
14-19, 2014.Comment: Discrete Mathematics, 201
Cycles of free words in several independent random permutations with restricted cycle lengths
In this text, we consider random permutations which can be written as free
words in several independent random permutations: firstly, we fix a non trivial
word in letters , secondly, for all , we
introduce a -tuple of independent random permutations
of , and the random permutation we are going to
consider is the one obtained by replacing each letter in by .
For example, for , . Moreover, we restrict the set of possible lengths of
the cycles of the 's: we fix sets of positive integers
and suppose that for all , for all , is uniformly distributed on
the set of permutations of which have all their cycle lengths in
. For all positive integer , we are going to give asymptotics, as
goes to infinity, on the number of cycles of length of
. We shall also consider the joint distribution of the random vectors
. We first prove that the order of in a
certain quotient of the free group with generators determines
the rate of growth of the random variables as goes to
infinity. We also prove that in many cases, the distribution of
converges to a Poisson law with parameter and that the random variables
are asymptotically independent. We notice
the surprising fact that from this point of view, many things happen as if
were uniformly distributed on the -th symmetric group.Comment: 28 page
Decompositions of edge-colored infinite complete graphs into monochromatic paths
An -edge coloring of a graph or hypergraph is a map . Extending results of Rado and answering questions of Rado,
Gy\'arf\'as and S\'ark\"ozy we prove that
(1.) the vertex set of every -edge colored countably infinite complete
-uniform hypergraph can be partitioned into monochromatic tight paths
with distinct colors (a tight path in a -uniform hypergraph is a sequence of
distinct vertices such that every set of consecutive vertices forms an
edge),
(2.) for all natural numbers and there is a natural number such
that the vertex set of every -edge colored countably infinite complete graph
can be partitioned into monochromatic powers of paths apart from a
finite set (a power of a path is a sequence of
distinct vertices such that implies that is an
edge),
(3.) the vertex set of every -edge colored countably infinite complete
graph can be partitioned into monochromatic squares of paths, but not
necessarily into ,
(4.) the vertex set of every -edge colored complete graph on
can be partitioned into monochromatic paths with distinct colors
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