48 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
Ramsey numbers of Berge-hypergraphs and related structures
For a graph , a hypergraph is called a Berge-,
denoted by , if there exists a bijection such
that for every , . Let the Ramsey number
be the smallest integer such that for any -edge-coloring of
a complete -uniform hypergraph on vertices, there is a monochromatic
Berge- subhypergraph. In this paper, we show that the 2-color Ramsey number
of Berge cliques is linear. In particular, we show that for and where is a Berge-
hypergraph. For higher uniformity, we show that for
and for and sufficiently large. We
also investigate the Ramsey number of trace hypergraphs, suspension hypergraphs
and expansion hypergraphs.Comment: Updated to include suggestions of the refere
Colored complete hypergraphs containing no rainbow Berge triangles
The study of graph Ramsey numbers within restricted colorings, in particular forbidding a rainbow triangle, has recently been blossoming under the name Gallai-Ramsey numbers. In this work, we extend the main structural tool from rainbow triangle free colorings of complete graphs to rainbow Berge triangle free colorings of hypergraphs. In doing so, some other concepts and results are also translated from graphs to hypergraphs
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
Ramsey Problems for Berge Hypergraphs
For a graph G, a hypergraph is a Berge copy of G (or a Berge-G in short) if there is a bijection such that for each we have . We denote the family of r-uniform hypergraphs that are Berge copies of G by . For families of r-uniform hypergraphs and , we denote by the smallest number n such that in any red-blue coloring of the (hyper)edges of (the complete r-uniform hypergraph on n vertices) there is a monochromatic blue copy of a hypergraph in or a monochromatic red copy of a hypergraph in . denotes the smallest number n such that in any coloring of the hyperedges of with c colors, there is a monochromatic copy of a hypergraph in . In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if , then . In the case r = 2c, we show that , and if G is a noncomplete graph on n vertices, then , assuming n is large enough. In the case we also obtain bounds on . Moreover, we also determine the exact value of for every pair of trees T_1 and T_2.
Read More: https://epubs.siam.org/doi/abs/10.1137/18M1225227?journalCode=sjdme