33 research outputs found
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Rainbow Generalizations of Ramsey Theory - A Dynamic Survey
In this work, we collect Ramsey-type results concerning rainbow edge colorings of graphs
Extremal and probabilistic results for regular graphs
In this thesis we explore extremal graph theory, focusing on new methods which apply to different notions of regular graph. The first notion is dregularity, which means each vertex of a graph is contained in exactly d edges, and the second notion is Szemerédi regularity, which is a strong, approximate version of this property that relates to pseudorandomness.
We begin with a novel method for optimising observables of Gibbs distributions in sparse graphs. The simplest application of the method is to the hard-core model, concerning independent sets in d-regular graphs, where we prove a tight upper bound on an observable known as the occupancy fraction. We also cover applications to matchings and colourings, in each case proving a tight bound on an observable of a Gibbs distribution and deriving an extremal result on the number of a relevant combinatorial structure in regular graphs. The results relate to a wide range of topics including statistical physics and Ramsey theory.
We then turn to a form of Szemerédi regularity in sparse hypergraphs, and develop a method for embedding complexes that generalises a widely-applied method for counting in pseudorandom graphs. We prove an inheritance lemma which shows that the neighbourhood of a sparse, regular subgraph
of a highly pseudorandom hypergraph typically inherits regularity in a natural way. This shows that we may embed complexes into suitable regular hypergraphs vertex-by-vertex, in much the same way as one can prove a counting lemma for regular graphs.
Finally, we consider the multicolour Ramsey number of paths and even cycles. A well-known density argument shows that when the edges of a complete graph on kn vertices are coloured with k colours, one can find a monochromatic path on n vertices. We give an improvement to this bound by exploiting the structure of the densest colour, and use the regularity method to extend the result to even cycles
Measure-Driven Algorithm Design and Analysis: A New Approach for Solving NP-hard Problems
NP-hard problems have numerous applications in various fields such as networks,
computer systems, circuit design, etc. However, no efficient algorithms have
been found for NP-hard problems. It has been commonly believed that no efficient algorithms
for NP-hard problems exist, i.e., that P6=NP. Recently, it has been observed
that there are parameters much smaller than input sizes in many instances of NP-hard
problems in the real world. In the last twenty years, researchers have been interested
in developing efficient algorithms, i.e., fixed-parameter tractable algorithms, for those
instances with small parameters. Fixed-parameter tractable algorithms can practically
find exact solutions to problem instances with small parameters, though those
problems are considered intractable in traditional computational theory.
In this dissertation, we propose a new approach of algorithm design and analysis:
discovering better measures for problems. In particular we use two measures instead of
the traditional single measure?input size to design algorithms and analyze their time
complexity. For several classical NP-hard problems, we present improved algorithms
designed and analyzed with this new approach,
First we show that the new approach is extremely powerful for designing fixedparameter
tractable algorithms by presenting improved fixed-parameter tractable algorithms
for the 3D-matching and 3D-packing problems, the multiway cut problem, the feedback vertex set problems on both directed and undirected
graph and the max-leaf problems on both directed and undirected graphs. Most of
our algorithms are practical for problem instances with small parameters.
Moreover, we show that this new approach is also good for designing exact algorithms
(with no parameters) for NP-hard problems by presenting an improved exact
algorithm for the well-known satisfiability problem.
Our results demonstrate the power of this new approach to algorithm design and
analysis for NP-hard problems. In the end, we discuss possible future directions on
this new approach and other approaches to algorithm design and analysis
Q(sqrt(-3))-Integral Points on a Mordell Curve
We use an extension of quadratic Chabauty to number fields,recently developed by the author with Balakrishnan, Besser and M ̈uller,combined with a sieving technique, to determine the integral points overQ(√−3) on the Mordell curve y2 = x3 − 4