74 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
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
- …