Complexity and sturmian colorings of regular trees

In graph theorygraph coloring is a special case of graph labeling ; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices of a graph such that no two adjacent vertices are of the same color; this is called a vertex coloring. Similarly, an edge coloring assigns a color to each edge so that no two adjacent edges are of the same color, and a face coloring of a planar graph assigns a color to each face or region so that no two faces that share a boundary have the same color.

Vertex coloring is usually used to introduce graph coloring problems, since other coloring problems can be transformed into a vertex coloring instance. For example, an edge coloring of a graph is just a vertex coloring of its line graphand a face coloring of a plane graph is just a vertex coloring of its dual. However, non-vertex coloring problems are often stated and studied as-is.

This is partly pedagogicaland partly because some problems are best studied in their non-vertex form, as in the case of edge coloring. The convention of using colors originates from coloring the countries of a mapwhere each face is literally colored.

This was generalized to coloring the faces of a graph embedded in the plane. By planar duality it became coloring the vertices, and in this form it generalizes to all graphs. In mathematical and computer representations, it is typical to use the first few positive or non-negative integers as the "colors". In general, one can use any finite set as the "color set". The nature of the coloring problem depends on the number of colors but not on what they are. Graph coloring enjoys many practical applications as well as theoretical challenges.

Beside the classical types of problems, different limitations can also be set on the graph, or on the way a color is assigned, or even on the color itself. It has even reached popularity with the general public in the form of the popular number puzzle Sudoku. Graph coloring is still a very active field of research. Note: Many terms used in this article are defined in Glossary of graph theory. The first results about graph coloring deal almost exclusively with planar graphs in the form of the coloring of maps.

While trying to color a map of the counties of England, Francis Guthrie postulated the four color conjecturenoting that four colors were sufficient to color the map so that no regions sharing a common border received the same color. Arthur Cayley raised the problem at a meeting of the London Mathematical Society in The same year, Alfred Kempe published a paper that claimed to establish the result, and for a decade the four color problem was considered solved.

Seonhee Lim

However, in that paper he proved the five color theoremsaying that every planar map can be colored with no more than five colors, using ideas of Kempe. In the following century, a vast amount of work and theories were developed to reduce the number of colors to four, until the four color theorem was finally proved in by Kenneth Appel and Wolfgang Haken.

The proof went back to the ideas of Heawood and Kempe and largely disregarded the intervening developments. InGeorge David Birkhoff introduced the chromatic polynomial to study the coloring problems, which was generalised to the Tutte polynomial by Tutteimportant structures in algebraic graph theory. Kempe had already drawn attention to the general, non-planar case in[3] and many results on generalisations of planar graph coloring to surfaces of higher order followed in the early 20th century.

InClaude Berge formulated another conjecture about graph coloring, the strong perfect graph conjectureoriginally motivated by an information-theoretic concept called the zero-error capacity of a graph introduced by Shannon. The conjecture remained unresolved for 40 years, until it was established as the celebrated strong perfect graph theorem by ChudnovskyRobertsonSeymourand Thomas in One of the major applications of graph coloring, register allocation in compilers, was introduced in A coloring of a graph is convex if it induces a partition of the vertices into connected subgraphs.

Besides being an interesting property from a theoretical point of view, tests for convexity have applications in various areas involving large graphs. We study the important subcase of testing for convexity in trees. This problem is linked, among other possible applications, with the study of phylogenetic trees, which are central in genetic research, and are used in linguistics and other areas. We also consider whether the dependency on k can be reduced in some cases, and provide an alternative testing algorithm for the case of paths.

Then we investigate a variant of convexity, namely quasi-convexity, in which all but one of the colors are required to induce connected components. For both our convexity and quasi-convexity tests, we show that, assuming that a query takes constant time, the time complexity can be reduced to a constant independent of n if we allow a preprocessing stage of time O n and O n2respectively.

Finally, we show how to test for a variation of convexity and quasi-convexity where the maximum number of connectivity classes of each color is allowed to be a constant value other than 1. Documents: Advanced Search Include Citations. Venue: Proc. Abstract A coloring of a graph is convex if it induces a partition of the vertices into connected subgraphs. Powered by:.Full-text: Access denied no subscription detected We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber.

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complexity and sturmian colorings of regular trees

If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text. The mixing time of the Glauber dynamics for spin systems on trees is closely related to the reconstruction problem. Martinelli, Sinclair and Weitz established this correspondence for a class of spin systems with soft constraints bounding the log-Sobolev constant by a comparison with the block dynamics [ Comm.

However, when there are hard constraints, the dynamics inside blocks may be reducible. We introduce a variant of the block dynamics extending these results to a wide class of spin systems with hard constraints.

complexity and sturmian colorings of regular trees

This applies to essentially any spin system that has nonreconstruction provided that on average the root is not locally frozen in a large neighborhood. Source Ann. Zentralblatt MATH identifier Subjects Primary: 60J Markov chains discrete-time Markov processes on discrete state spaces. Keywords Mixing time Glauber dynamics graph colorings reconstruction problem.

Sly, Allan; Zhang, Yumeng. The Glauber dynamics of colorings on trees is rapidly mixing throughout the nonreconstruction regime. Read more about accessing full-text Buy article. Abstract Article info and citation First page References Abstract The mixing time of the Glauber dynamics for spin systems on trees is closely related to the reconstruction problem. Article information Source Ann.

Export citation. Export Cancel. References [1] Berger, N.

Micros

Glauber dynamics on trees and hyperbolic graphs. Theory Related Fields — Reconstruction for colorings on trees. SIAM J. Discrete Math.

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The Kesten—Stigum reconstruction bound is tight for roughly symmetric binary channels. Mixing time of critical Ising model on trees is polynomial in the height. A survey on the use of Markov chains to randomly sample colourings.On the multiple recurrence properties for disjoint systems, preprint. Systemsto appear. IMRNto appear. Korean Math. IMRN, Systems39 Fourier67 Pisa Cl.

Systems35 Systems34 Refined shrinking target property of rotations, Nonlinearity27 ID6 pp. Metric inhomogeneous diophantine approximation on the field of formal Laurent series, Acta Arith. Quantitative recurrence properties for group actions, Nonlinearity22 The recurrence time for interval exchange maps, Nonlinearity21 A new version of first return time test of pseudorandomness, J.

KSIAM12— Tests of randomness by the gambler's ruin algorithm, Appl. The dynamical Borel-Cantelli lemma for interval maps, Discrete Contin. The recurrence time for ergodic systems with infinite invariant measures, Nonlinearity 19 On the law of logarithm of the recurrence time, Discrete Contin. The first return time test for pseudorandom numbers, J.

Papers On the multiple recurrence properties for disjoint systems, preprint.We study colorings of a tree induced from isometries of the hyperbolic plane given an ideal tessellation.

We show that, for a given tessellation of the hyperbolic plane by ideal polygons, a coloring can be associated with any element of Isomand the element is a commensurator of if and only if its associated coloring is periodic, generalizing a result of Hedlund and Morse. Let be a locally finite tree, its vertex set, and the set of oriented edges of. Let be a countable set which will be called the alphabet.

Let be a coloring ofthat is, a map. Let be the automorphism group of. A periodic coloring is a coloring which is -invariant for some cocompact subgroup. In this paper, we study colorings of regular trees induced from some tessellations of the hyperbolic plane. There is a well-known family of sequences coming from rotations of circle as follows.

Consider the tiling of the real line by unit length intervals and a map from to itself. There exists an integer such that each interval is partitioned into or subintervals of the form. Consider the sequence withwhich is given by the number of such subintervals of.

complexity and sturmian colorings of regular trees

It is well known that this two-sided sequence is periodic if only if is rational [ 1 ]. As a generalization, we associate a coloring of a -regular tree for any isometry of the hyperbolic plane, given a specific hyperbolic tessellation generated by a discrete subgroup of the group of isometries on the hyperbolic plane.

Suppose that each vertex of elements of lies on the boundary of the hyperbolic plane so that the dual graph of is a tree. For such a tessellationwe show that the coloring is periodic if and only if is a commensurator of in. Recall that an element is called a commensurator of if and only if is a subgroup of and of of finite index.

Let us denote the group of commensurators of by. Commensurator subgroup plays an important role in the study of rigidity of locally symmetric spaces and more generally in geometric group theory [ 2 — 4 ]. This is a result analogous to the rotation case in the sense that the group of commensurators of is a group containing with finite index [ 5 ].

After showing the main theorem Theorem 3we show that our construction is an analogue of sequences induced from a rotation of circle only when the multiplicative constant of is rational. We show that, in the case of an isometry of which is not a commensurator, we obtain colorings of unbounded alphabet, in contrast with the motivating example where irrational rotations correspond to Sturmian sequences, which are in particular sequences with a finite alphabet see Section 3 for details.

We first reformulate the classical example of two-sided sequences mentioned in Section 1. Consider the tessellation of the hyperbolic plane upper-half plane given by the group generated by the reflections about the lines and. More precisely, elements of are of the form. Then is isomorphic to the infinite dihedral group, and its dual graph is a 2-regular tree.

Letwhich sends to. Then it is not difficult to check that is a commensurator of if and only ifand is rational. For each vertexdenote by the element of dual to. Let be the coloring given by Then is periodic if is rational,and Sturmian if is irrational, e. Let us generalize the above construction. Let us fix an ideal polygon in the hyperbolic plane. Consider the group generated by the reflections in the edges ofwhich is a discrete subgroup of finite covolume in the isometry group of.

Let be the dual graph of the tessellationwhich is a tree since is an ideal polygon. The tree is the Cayley graph of the group.Nevertheless, there is never a. I'll discuss this and alsi some other examples of these phenomena. This is joint work with Benjamin Weiss and Zemer Kosloff. Abstract: Separated nets a.

Graph coloring

Delone sets can be constructed from R d actions on the torus, via the cut-and-project method. This method produces examples of aperiodic patterns which occur in nature e.

In previous joint work we showed that generically, cut-and-project nets are bi-Lipschitz equivalent to a lattice, and that, for some choices of dimensions, they are generically bounded distance to a lattice. The goal of this talk is to explain a recent proof that, in any co-dimension one cut-and-project setup, regardless of Diophantine properties, the acceptance domain can always be chosen in a non-trivial way so that the resulting separated net is bounded distance to a lattice.

Abstract: We construct new counterexamples to the Lagarias-Wang finiteness conjecture on the joint spectral radius. The approach uses ergodic optimization and Sturmian measures.

This is joint work with Mark Pollicott. Abstract: In this talk, we introduce subword complexity of colorings of regular trees. We characterize colorings of bounded subword complexity and then introduce Sturmian colorings, which are colorings of minimal unbounded subword complexity.

We classify Sturmian colorings using their type sets. We show that any Sturmian coloring is a lifting of a coloring on a quotient graph of the tree which is a geodesic or a ray, with loops possibly attached, thus a lifting of an ''infinte word". We further give a complete characterization of the quotient graph for eventually periodic ones. We will provide several examples. This is joint work with Seonhee Lim.

Title: A class of inhomogeneous Markov shifts without an absolutely continuous invariant measure. Abstract: An inhomogeneous Markov shift is the shift of a Markov chain where the transitions may vary over time.

Topological Markov shifts are a class of symbolic topological dynamical systems which play a crucial role in the study of dynamical systems arising from differential equations. Abstract: We introduce an operator renewal equation for dynamical systems with continuous time. Using this equation, we prove results on mixing and rates of mixing for both finite and infinite measure flows.

This is joint work with Dalia Terhesiu.

complexity and sturmian colorings of regular trees

Abstract: The goal of this talk is to bring to light some recently discovered connections between problems about graph colourings and problems about the approximation of real numbers by rationals. The connections arise as a result of the fact that many statements about the quality of approximation of real numbers can be phrased as problems about the orbits of points in certain spaces e.

Once these group actions are identified, there is a correspondence between questions about the approximations and questions about the Cayley graph of the given group. Information on either side of this correspondence gives information about the other.Bezos accurately predicted what Amazon would grow into. Customers buy everything from clothes to groceries to kitchen appliances from the e-commerce giant.

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