270 research outputs found
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
Ramsey numbers of ordered graphs
An ordered graph is a pair where is a graph and
is a total ordering of its vertices. The ordered Ramsey number
is the minimum number such that every ordered
complete graph with vertices and with edges colored by two colors contains
a monochromatic copy of .
In contrast with the case of unordered graphs, we show that there are
arbitrarily large ordered matchings on vertices for which
is superpolynomial in . This implies that
ordered Ramsey numbers of the same graph can grow superpolynomially in the size
of the graph in one ordering and remain linear in another ordering.
We also prove that the ordered Ramsey number is
polynomial in the number of vertices of if the bandwidth of
is constant or if is an ordered graph of constant
degeneracy and constant interval chromatic number. The first result gives a
positive answer to a question of Conlon, Fox, Lee, and Sudakov.
For a few special classes of ordered paths, stars or matchings, we give
asymptotically tight bounds on their ordered Ramsey numbers. For so-called
monotone cycles we compute their ordered Ramsey numbers exactly. This result
implies exact formulas for geometric Ramsey numbers of cycles introduced by
K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of
Combinatoric
Practical and theoretical applications of the Regularity Lemma
The Regularity Lemma of Szemeredi is a fundamental tool in extremal graph theory with a wide range of applications in theoretical computer science. Partly as a recognition of his work on the Regularity Lemma, Endre Szemeredi has won the Abel Prize in 2012 for his outstanding achievement. In this thesis we present both practical and theoretical applications of the Regularity Lemma. The practical applications are concerning the important problem of data clustering, the theoretical applications are concerning the monochromatic vertex partition problem. In spite of its numerous applications to establish theoretical results, the Regularity Lemma has a drawback that it requires the graphs under consideration to be astronomically large, thus limiting its practical utility. As stated by Gowers, it has been ``well beyond the realms of any practical applications\u27, the existing applications have been theoretical, mathematical. In the first part of the thesis, we propose to change this and we propose some modifications to the constructive versions of the Regularity Lemma. While this affects the generality of the result, it also makes it more useful for much smaller graphs. We call this result the practical regularity partitioning algorithm and the resulting clustering technique Regularity Clustering. This is the first integrated attempt in order to make the Regularity Lemma applicable in practice. We present results on applying regularity clustering on a number of benchmark data-sets and compare the results with k-means clustering and spectral clustering. Finally we demonstrate its application in Educational Data Mining to improve the student performance prediction. In the second part of the thesis, we study the monochromatic vertex partition problem. To begin we briefly review some related topics and several proof techniques that are central to our results, including the greedy and absorbing procedures. We also review some of the current best results before presenting ours, where the Regularity Lemma has played a critical role. Before concluding we discuss some future research directions that appear particularly promising based on our work
Partitioning a 2-edge-coloured graph of minimum degree into three monochromatic cycles
Lehel conjectured in the 1970s that every red and blue edge-coloured complete
graph can be partitioned into two monochromatic cycles. This was confirmed in
2010 by Bessy and Thomass\'e. However, the host graph does not have to be
complete. It it suffices to require that has minimum degree at least
, where is the order of , as was shown recently by Letzter,
confirming a conjecture of Balogh, Bar\'{a}t, Gerbner, Gy\'arf\'as and
S\'ark\"ozy. This degree condition is asymptotically tight.
Here we continue this line of research, by proving that for every red and
blue edge-colouring of an -vertex graph of minimum degree at least , there is a partition of the vertex set into three monochromatic cycles.
This approximately verifies a conjecture of Pokrovskiy and is essentially
tight
Large monochromatic components in expansive hypergraphs
A result of Gy\'arf\'as exactly determines the size of a largest
monochromatic component in an arbitrary -coloring of the complete
-uniform hypergraph when and . We prove a
result which says that if one replaces in Gy\'arf\'as' theorem by any
``expansive'' -uniform hypergraph on vertices (that is, a -uniform
hypergraph on vertices in which in which for all
disjoint sets with for all ), then one gets a largest monochromatic component of essentially the same
size (within a small error term depending on and ). As corollaries
we recover a number of known results about large monochromatic components in
random hypergraphs and random Steiner triple systems, often with drastically
improved bounds on the error terms.
Gy\'arf\'as' result is equivalent to the dual problem of determining the
smallest maximum degree of an arbitrary -partite -uniform hypergraph with
edges in which every set of edges has a common intersection. In this
language, our result says that if one replaces the condition that every set of
edges has a common intersection with the condition that for every
collection of disjoint sets with
for all there exists for all
such that , then the maximum degree of
is essentially the same (within a small error term depending on and
). We prove our results in this dual setting.Comment: 18 page
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
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