18 research outputs found
Partitioning 3-colored complete graphs into three monochromatic cycles
We show in this paper that in every 3-coloring of the edges of Kn all but o(n)
of its vertices can be partitioned into three monochromatic cycles. From this, using
our earlier results, actually it follows that we can partition all the vertices into at
most 17 monochromatic cycles, improving the best known bounds. If the colors of
the three monochromatic cycles must be different then one can cover ( 3
4 â o(1))n
vertices and this is close to best possible
Problems and memories
I state some open problems coming from joint work with Paul Erd\H{o}sComment: This is a paper form of the talk I gave on July 5, 2013 at the
centennial conference in Budapest to honor Paul Erd\H{o}
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
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
Coverings by Few Monochromatic Pieces: A Transition Between Two Ramsey Problems
The typical problem in (generalized) Ramsey theory is to find the order of the largest monochromatic member of a family {Mathematical expression} (for example matchings, paths, cycles, connected subgraphs) that must be present in any edge coloring of a complete graph Kn with t colors. Another area is to find the minimum number of monochromatic members of {Mathematical expression} that partition or cover the vertex set of every edge colored complete graph. Here we propose a problem that connects these areas: for a fixed positive integers s ⤠t, at least how many vertices can be covered by the vertices of no more than s monochromatic members of {Mathematical expression} in every edge coloring of Kn with t colors. Several problems and conjectures are presented, among them a possible extension of a well-known result of Cockayne and Lorimer on monochromatic matchings for which we prove an initial step: every t-coloring of Kn contains a (t - 1)-colored matching of size k provided that {Mathematical expression} Š 2013 Springer Japan
Monochromatic cycle partitions in random graphs
Erd\H{o}s, Gy\'arf\'as and Pyber showed that every -edge-coloured complete
graph can be covered by vertex-disjoint monochromatic
cycles (independent of ). Here, we extend their result to the setting of
binomial random graphs. That is, we show that if , then with high probability any -edge-coloured
can be covered by at most vertex-disjoint monochromatic
cycles. This answers a question of Kor\'andi, Mousset, Nenadov, \v{S}kori\'{c}
and Sudakov.Comment: 16 pages, accepted in Combinatorics, Probability and Computin