83 research outputs found
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
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
Partitioning infinite hypergraphs into few monochromatic Berge-paths
Extending a result of Rado to hypergraphs, we prove that for all s, k, t∈ N with k≥ t≥ 2 , the vertices of every r= s(k- t+ 1) -edge-coloured countably infinite complete k-graph can be partitioned into the cores of at most s monochromatic t-tight Berge-paths of different colours. We further describe a construction showing that this result is best possible
Partitioning Edge-Colored Hypergraphs into Few Monochromatic Tight Cycles
Confirming a conjecture of Gy´arf´as, we prove that, for all natural numbers k and
r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a
bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We
further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete
graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes,
Soukup, Soukup, and Szentmikl´ossy [Discrete Math., 340 (2017), pp. 2053–2069]. In fact we prove a
common generalization of both theorems which further extends these results to all host hypergraphs
of bounded independence number
Partitioning edge-colored hypergraphs into few monochromatic tight cycles
Confirming a conjecture of Gyárfás, we prove that, for all natural numbers k and r, the vertices of every r-edge-colored complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r, the vertices of every r-edge-colored complete graph can be partitioned into a bounded number of pth powers of cycles, settling a problem of Elekes, Soukup, Soukup, and Szentmiklóssy [Discrete Math., 340 (2017), pp. 2053-2069]. In fact we prove a common generalization of both theorems which further extends these results to all host hypergraphs of bounded independence number
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