193 research outputs found
Local colourings and monochromatic partitions in complete bipartite graphs
We show that for any -local colouring of the edges of the balanced
complete bipartite graph , its vertices can be covered with at
most~ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite -locally coloured graph with
disjoint monochromatic cycles.\\ We also determine the -local
bipartite Ramsey number of a path almost exactly: Every -local colouring of
the edges of contains a monochromatic path on vertices.Comment: 18 page
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
Partitioning edge-coloured complete graphs into monochromatic cycles and paths
A conjecture of Erd\H{o}s, Gy\'arf\'as, and Pyber says that in any
edge-colouring of a complete graph with r colours, it is possible to cover all
the vertices with r vertex-disjoint monochromatic cycles. So far, this
conjecture has been proven only for r = 2. In this paper we show that in fact
this conjecture is false for all r > 2. In contrast to this, we show that in
any edge-colouring of a complete graph with three colours, it is possible to
cover all the vertices with three vertex-disjoint monochromatic paths, proving
a particular case of a conjecture due to Gy\'arf\'as. As an intermediate result
we show that in any edge-colouring of the complete graph with the colours red
and blue, it is possible to cover all the vertices with a red path, and a
disjoint blue balanced complete bipartite graph.Comment: 25 pages, 3 figure
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
Vertex covering with monochromatic pieces of few colours
In 1995, Erd\H{o}s and Gy\'arf\'as proved that in every -colouring of the
edges of , there is a vertex cover by monochromatic paths of
the same colour, which is optimal up to a constant factor. The main goal of
this paper is to study the natural multi-colour generalization of this problem:
given two positive integers , what is the smallest number
such that in every colouring of the edges of with
colours, there exists a vertex cover of by
monochromatic paths using altogether at most different colours? For fixed
integers and as , we prove that , where is the chromatic number of
the Kneser gr aph . More generally, if one replaces by
an arbitrary -vertex graph with fixed independence number , then we
have , where this time around is the
chromatic number of the Kneser hypergraph . This
result is tight in the sense that there exist graphs with independence number
for which . This is in sharp
contrast to the case , where it follows from a result of S\'ark\"ozy
(2012) that depends only on and , but not on
the number of vertices. We obtain similar results for the situation where
instead of using paths, one wants to cover a graph with bounded independence
number by monochromatic cycles, or a complete graph by monochromatic
-regular graphs
Monochromatic cycle partitions in local edge colourings
An edge colouring of a graph is said to be an -local colouring if the
edges incident to any vertex are coloured with at most colours.
Generalising a result of Bessy and Thomass\'e, we prove that the vertex set of
any -locally coloured complete graph may be partitioned into two disjoint
monochromatic cycles of different colours. Moreover, for any natural number
, we show that the vertex set of any -locally coloured complete graph may
be partitioned into disjoint monochromatic cycles. This
generalises a result of Erd\H{o}s, Gy\'arf\'as and Pyber.Comment: 10 page
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