533 research outputs found
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on
a version of the triangular lattice in the complex plane have unique scaling
limits, which are invariant under rotations, scalings, and, in the case of the
MST, also under translations. However, they are not expected to be conformally
invariant. We also prove some geometric properties of the limiting MST. The
topology of convergence is the space of spanning trees introduced by Aizenman,
Burchard, Newman & Wilson (1999), and the proof relies on the existence and
conformal covariance of the scaling limit of the near-critical percolation
ensemble, established in our earlier works.Comment: 56 pages, 21 figures. A thoroughly revised versio
Equitable edge colored Steiner triple systems
A k-edge coloring of G is said to be equitable if the number of edges, at any vertex, colored with a certain color differ by at most one from the number of edges colored with a different color at the same vertex. An STS(v) is said to be polychromatic if the edges in each triple are colored with three different colors. In this paper, we show that every STS(v) admits a 3-edge coloring that is both polychromatic for the STS(v) and equitable for the underlying complete graph. Also, we show that, for v 1 or 3 (mod 6), there exists an equitable k-edge coloring of K which does not admit any polychromatic STS(v), for k = 3 and k = v - 2
Long rainbow cycles in proper edge-colorings of complete graphs
We show that any properly edge-colored Kn contains a rainbow cycle with
at least (4=7 − o(1))n edges. This improves the lower bound of n=2 − 1 proved
in [1]
An upper bound on geodesic length in 2D critical first-passage percolation
We consider i.i.d. first-passage percolation (FPP) on the two-dimensional
square lattice, in the critical case where edge-weights take the value zero
with probability . Critical FPP is unique in that the Euclidean lengths of
geodesics are superlinear, rather than linear, in the distance between their
endpoints. This fact was speculated by Kesten in 1986 but not confirmed until
2019 by Damron and Tang, who showed a lower bound on geodesic length that is
polynomial with degree strictly greater than . In this paper we establish
the first non-trivial upper bound. Namely, we prove that for a large class of
critical edge-weight distributions, the shortest geodesic from the origin to a
box of radius uses at most edges with high
probability, for any . Here is the polychromatic 3-arm
probability from classical Bernoulli percolation; upon inserting its
conjectural asymptotic, our bound converts to . In any
case, it is known that for some ,
and so our bound gives an exponent strictly less than . In the special case
of Bernoulli() edge-weights, we replace the additional factor of
with a constant and give an expectation bound.Comment: 62 pages, 14 figure
Extremal colorings and extremal satisfiability
Combinatorial problems are often easy to state and hard to solve. A whole bunch of graph coloring problems falls into this class as well as the satisfiability problem. The classical coloring problems consider colorings of objects such that two objects which are in a relation receive different colors, e.g., proper vertex-colorings, proper edge-colorings, or proper face-colorings of plane graphs.
A generalization is to color the objects such that some predefined patterns are not monochromatic. Ramsey theory deals with questions under what conditions such colorings can occur. A more restrictive version of colorings forces some substructures to be polychromatic, i.e., to receive all colors used in the coloring at least once. Also a true-false-assignment to the boolean variables of a formula can be seen as a 2-coloring of the literals where there are restrictions that complementary literals receive different colors.
Mostly, the hardness of such problems is been made explicit by proving that they are NP-hard. This indicates that there might be no simple characterization of all solvable instances. Extremal questions then become quite handy, because they do not aim at a complete characteriziation, but rather focus on one parameter and ask for its minimum or maximum value.
The goal of this thesis is to demonstrate this general way on different problems in the area of graph colorings and satisfiability of boolean formulas.
First, we consider graphs where all edge-2-colorings contain a monochromatic copy of some fixed graph H. Such graphs are called H-Ramsey graphs and we concentrate on their minimum degree. Its minimization is the question we are going to answer for H being a biregular bipartite graph, a forest, or a bipartite graph where the size of both partite sets are equal.
Second, vertex-colorings of plane multigraphs are studied such that each face is polychromatic. A natural parameter to upper bound the number of colors which can be used in such a coloring is the size g of the smallest face. We show that every graph can be polychromatically colored with \floor{3g-5}{4} colors and there are examples for which this bound is almost tight.
Third, we consider a variant of the satisfiability problem where only some (not necessarily all) assignments are allowed. A natural way to choose such a set of allowed assignments is to use a context-free language. If in addition the number of all allowed assignments of length n is lower bounded by (an) for some , then this restricted satisfiability problem will be shown to be NP-hard. Otherwise, there are only polynomially many allowed assignments and the restricted satisfiability problem is proven to be polynomially solvable
The scaling limits of the Minimal Spanning Tree and Invasion Percolation in the plane
We prove that the Minimal Spanning Tree and the Invasion Percolation Tree on a version of the triangular lattice in the complex plane have unique scaling limits, which are invariant under rotations, scalings, and, in the case of the MST, also under translations. However, they are not expected to be conformally invariant. We also prove some geometric properties of the limiting MST. The topology of convergence is the space of spanning trees introduced by Aizenman, Burchard, Newman & Wilson (1999), and the proof relies on the existence and conformal covariance of the scaling limit of the near-critical percolation ensemble, established in our earlier works
Planar percolation with a glimpse of Schramm-Loewner Evolution
In recent years, important progress has been made in the field of
two-dimensional statistical physics. One of the most striking achievements is
the proof of the Cardy-Smirnov formula. This theorem, together with the
introduction of Schramm-Loewner Evolution and techniques developed over the
years in percolation, allow precise descriptions of the critical and
near-critical regimes of the model. This survey aims to describe the different
steps leading to the proof that the infinite-cluster density for
site percolation on the triangular lattice behaves like
as .Comment: Survey based on lectures given in "La Pietra week in Probability",
Florence, Italy, 2011. (2013
- …