2,107 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
Covering graphs by monochromatic trees and Helly-type results for hypergraphs
How many monochromatic paths, cycles or general trees does one need to cover
all vertices of a given -edge-coloured graph ? These problems were
introduced in the 1960s and were intensively studied by various researchers
over the last 50 years. In this paper, we establish a connection between this
problem and the following natural Helly-type question in hypergraphs. Roughly
speaking, this question asks for the maximum number of vertices needed to cover
all the edges of a hypergraph if it is known that any collection of a few
edges of has a small cover. We obtain quite accurate bounds for the
hypergraph problem and use them to give some unexpected answers to several
questions about covering graphs by monochromatic trees raised and studied by
Bal and DeBiasio, Kohayakawa, Mota and Schacht, Lang and Lo, and Gir\~ao,
Letzter and Sahasrabudhe.Comment: 20 pages including references plus 2 pages of an Appendi
Hamiltonicity and -hypergraphs
We define and study a special type of hypergraph. A -hypergraph ), where is a partition of , is an
-uniform hypergraph having vertices partitioned into classes of
vertices each. If the classes are denoted by , ,...,, then a
subset of of size is an edge if the partition of formed by
the non-zero cardinalities , ,
is . The non-empty intersections are called the parts
of , and denotes the number of parts. We consider various types
of cycles in hypergraphs such as Berge cycles and sharp cycles in which only
consecutive edges have a nonempty intersection. We show that most
-hypergraphs contain a Hamiltonian Berge cycle and that, for and , a -hypergraph always contains a sharp
Hamiltonian cycle. We also extend this result to -intersecting cycles
- …