25,825 research outputs found
Covering the complete graph by partitions
AbstractA (D, c)-coloring of the complete graph Kn is a coloring of the edges with c colors such that all monochromatic connected subgraphs have at most D vertices. Resolvable block designs with c parallel classes and with block size D are natural examples of (D, c)-colorings. However, (D, c)-colorings are more relaxed structures. We investigate the largest n such that Kn has a (D, c)-coloring. Our main tool is the fractional matching theory of hypergraphs
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
Local colourings and monochromatic partitions in complete bipartite graphs
We show that for any -local colouring of the edges of the balanced
complete bipartite graph , its vertices can be covered with at
most~ disjoint monochromatic paths. And, we can cover almost all vertices of
any complete or balanced complete bipartite -locally coloured graph with
disjoint monochromatic cycles.\\ We also determine the -local
bipartite Ramsey number of a path almost exactly: Every -local colouring of
the edges of contains a monochromatic path on vertices.Comment: 18 page
Minimum degree conditions for monochromatic cycle partitioning
A classical result of Erd\H{o}s, Gy\'arf\'as and Pyber states that any
-edge-coloured complete graph has a partition into
monochromatic cycles. Here we determine the minimum degree threshold for this
property. More precisely, we show that there exists a constant such that
any -edge-coloured graph on vertices with minimum degree at least has a partition into monochromatic cycles. We also
provide constructions showing that the minimum degree condition and the number
of cycles are essentially tight.Comment: 22 pages (26 including appendix
Approximating Minimum Cost Connectivity Orientation and Augmentation
We investigate problems addressing combined connectivity augmentation and
orientations settings. We give a polynomial-time 6-approximation algorithm for
finding a minimum cost subgraph of an undirected graph that admits an
orientation covering a nonnegative crossing -supermodular demand function,
as defined by Frank. An important example is -edge-connectivity, a
common generalization of global and rooted edge-connectivity.
Our algorithm is based on a non-standard application of the iterative
rounding method. We observe that the standard linear program with cut
constraints is not amenable and use an alternative linear program with
partition and co-partition constraints instead. The proof requires a new type
of uncrossing technique on partitions and co-partitions.
We also consider the problem setting when the cost of an edge can be
different for the two possible orientations. The problem becomes substantially
more difficult already for the simpler requirement of -edge-connectivity.
Khanna, Naor, and Shepherd showed that the integrality gap of the natural
linear program is at most when and conjectured that it is constant
for all fixed . We disprove this conjecture by showing an
integrality gap even when
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