463 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
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
Embeddings and Ramsey numbers of sparse k-uniform hypergraphs
Chvatal, Roedl, Szemeredi and Trotter proved that the Ramsey numbers of
graphs of bounded maximum degree are linear in their order. In previous work,
we proved the same result for 3-uniform hypergraphs. Here we extend this result
to k-uniform hypergraphs, for any integer k > 3. As in the 3-uniform case, the
main new tool which we prove and use is an embedding lemma for k-uniform
hypergraphs of bounded maximum degree into suitable k-uniform `quasi-random'
hypergraphs.Comment: 24 pages, 2 figures. To appear in Combinatoric
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